Miguel Couceiro

Jimmy Devillet

Every quasitrivial n-ary semigroup is reducible to a semigroup

Keywords: Mathematics Subject Classification. 05A15, 20N15; 16B99, 20M14. Keywords. quasitrivial polyadic semigroup, reducibility, symmetry, enumeration

We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences.

Introduction

Let X be a nonempty set and let n ≥ 2 be an integer. In this paper we are interested in n-ary operations F ∶ X n → X that are associative, i.e., that satisfy the following system of identities F (x 1 , . . . , x i-1 , F (x i , . . . , x i+n-1 ), x i+n , . . . , x 2n-1 ) = F (x 1 , . . . , x i , F (x i+1 , . . . , x i+n ), x i+n+1 , . . . , x 2n-1 ) for all x 1 , . . . , x 2n-1 ∈ X and all 1 ≤ i ≤ n -1. This generalisation of associativity was originally proposed by Dörnte [START_REF] Dörnte | Untersuchungen über einen verallgemeinerten Gruppenbegriff[END_REF] and studied by Post [START_REF] Post | Polyadic groups[END_REF] in the framework of n-ary groups and their reductions. An operation F ∶ X n → X is said to be reducible to a binary operation (resp. ternary operation) if it can be written as a composition of a binary (resp. ternary) associative operation (see Definition 2).

Recently, the study of reducibility criteria for n-ary semigroups1 gained an increasing interest (see, e.g., [START_REF] Ackerman | A characterization of quasitrivial n-semigroups[END_REF][START_REF] Dudek | On n-ary semigroups with adjoint neutral element[END_REF][START_REF] Devillet | Characterizations of quasitrivial symmetric nondecreasing associative operations[END_REF][START_REF] Kiss | Associative idempotent nondecreasing functions are reducible[END_REF]). In particular, Dudek and Mukhin [START_REF] Dudek | On n-ary semigroups with adjoint neutral element[END_REF] provided necessary and sufficient conditions under which an n-ary associative operation is reducible to a binary associative operation. Indeed, they proved (see [5, Theorem 1]) that an associative operation F ∶ X n → X is reducible to an associative binary operation if and only if one can adjoin a neutral element e to X for F , that is, there is an n-ary associative extension F ∶ (X ⋃{e}) n → X ⋃{e} of F such that e is a neutral element for F and F X n = F . In this case, a binary reduction G e of F can be defined by G e (x, y) = F (x, e, . . . , e n-2 times , y) x, y ∈ X.

Recently, Ackerman [START_REF] Ackerman | A characterization of quasitrivial n-semigroups[END_REF] also investigated reducibility criteria for n-ary associative operations that are quasitrivial2 , i.e., operations that preserve all unary relations: for every x 1 , . . . , x n ∈ X, F (x 1 , . . . , x n ) ∈ {x 1 , . . . , x n }.

The following result reassembles Corollaries 3.14 and 3.15, and Theorem 3.18 of [START_REF] Ackerman | A characterization of quasitrivial n-semigroups[END_REF].

Theorem 1. Let F ∶ X n → X be an associative and quasitrivial operation.

(a) If n is even, then F is reducible to an associative and quasitrivial binary operation G∶ X 2 → X. (b) If n is odd, then F is reducible to an associative and quasitrivial ternary operation H∶ X 3 → X. (c) If n = 3 and F is not reducible to an associative binary operation G∶ X 2 → X, then there exist a 1 , a 2 ∈ X with a 1 ≠ a 2 such that • F (X∖{a1,a2}) 3 is reducible to an associative binary operation.

• a 1 and a 2 are neutral elements for F .

From Theorem 1 (c) it would follow that if an associative and quasitrivial operation F ∶ X n → X is not reducible to an associative binary operation G∶ X 2 → X, then n is odd and there exist distinct a 1 , a 2 ∈ X that are neutral elements for F .

However, Theorem 1 (c) supposes the existence of a ternary associative and quasitrivial operation H∶ X 3 → X that is not reducible to an associative binary operation, and Ackerman did not provide any example of such an operation.

In this paper we show that there is no associative and quasitrivial n-ary operation that is not reducible to an associative binary operation (Corollary 4). Hence, for any associative and quasitrivial operation F ∶ X n → X one can adjoin a neutral element to X. Now this raises the question of whether such a binary reduction is unique and whether it is quasitrivial. We show that both of these properties are equivalent to the existence of at most one neutral element for the n-ary associative and quasitrivial operation (Theorem 18). Since an n-ary associative and quasitrivial operation has at most one neutral element when n is even or at most two when n is odd (Proposition 16), in the case when X is finite, we also provide several enumeration results (Propositions 22 and 24) that explicitly determine the sizes of the corresponding classes of associative and quasitrival n-ary operations in terms of the size of the underlying set X. As a by-product, these enumeration results led to several integer sequences that were previously unknown in the Sloane's On-Line Encyclopedia of Integer Sequences (OEIS, see [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF]). These results are further refined in the case of symmetric operations (Theorem 30).

Motivating results

In this section we recall some basic definitions and present some motivating results. In particular, we show that every associative and quasitrivial operation F ∶ X n → X is reducible to an associative binary operation (Corollary 4).

Throughout this paper let k ≥ 1 and x ∈ X. We use the shorthand notation [k] = {1, . . . , k} and n ⋅ x = x, . . . , x (n times), and we denote the set of all constant n-tuples over X by ∆ n X = {(n ⋅ y) y ∈ X}. Also, we denote the size of any set S by S .

Recall that a neutral element for F

∶ X n → X is an element e F ∈ X such that F ((i -1) ⋅ e F , x, (n -i) ⋅ e F ) = x for all x ∈ X and all i ∈ [n].
When the meaning is clear from the context, we may drop the index F and denote a neutral element for F by e.

Definition 2 (see [START_REF] Ackerman | A characterization of quasitrivial n-semigroups[END_REF][START_REF] Dudek | On n-ary semigroups with adjoint neutral element[END_REF]). Let G∶ X 2 → X, and H∶ X 3 → X be associative operations.

(1) An operation F ∶ X n → X is said to be reducible to

G if F (x 1 , . . . , x n ) = G n-1 (x 1 , . . . , x n ) for all x 1 , . . . , x n ∈ X, where G 1 = G and G m (x 1 , . . . , x m+1 ) = G m-1 (x 1 , . . . , x m-1 , G(x m , x m+1 )), for each 2 ≤ m ≤ n-1.
In this case, G is said to be a binary reduction of F . (2) Similarly, F is said to be reducible to H if n is odd and

F (x 1 , . . . , x n ) = H n-3 (x 1 , . . . , x n ) for all x 1 , . . . , x n ∈ X, where H 0 = H and H m (x 1 , . . . , x m+3 ) = H m-2 (x 1 , . . . , x m , H(x m+1 , x m+2 , x m+3 )),
for each even integer 2 ≤ m ≤ n -3. In this case, H is said to be a ternary reduction of F .

As we will see, every associative and quasitrivial operation F ∶ X n → X is reducible to an associative binary operation. To show this, we will make use of the follwing auxiliary result.

Lemma 3 (see [START_REF] Dudek | On n-ary semigroups with adjoint neutral element[END_REF]Lemma 1]). If F ∶ X n → X is associative and has a neutral element e ∈ X, then F is reducible to the associative operation G e ∶ X 2 → X defined by G e (x, y) = F (x, (n -2) ⋅ e, y), for every x, y ∈ X.

(

) 1
The following corollary follows from Theorem 1 and Lemma 3.

Corollary 4. Every associative and quasitrivial operation F ∶ X n → X is reducible to an associative binary operation.

Theorem 1(c) states that a ternary associative and quasitrivial operation F ∶ X 3 → X must have two neutral elements, whenever it is not reducible to a binary operation. In particular, we can show that two distinct elements a 1 , a 2 ∈ X are neutral elements for F if and only if they are neutral elements for the restriction F {a1,a2} 3 of F to {a 1 , a 2 } 3 . Indeed, the condition is obviously necessary, while its sufficiency follows from the Lemma 5 below. Lemma 5. Let H∶ X 3 → X be an associative and quasitrivial operation.

(a) If a 1 , a 2 ∈ X are two distinct neutral elements for H {a1,a2} 3 , then

H(a 1 , a 1 , x) = H(x, a 1 , a 1 ) = x = H(x, a 2 , a 2 ) = H(a 2 , a 2 , x), x ∈ X.
(b) If a 1 , a 2 ∈ X are two distinct neutral elements for H {a1,a2} 3 , then both a 1 and a 2 are neutral elements for H.

Proof. (a) Let x ∈ X. We only show that H(a 1 , a 1 , x) = x, since the other equalities can be shown similarly. Clearly, the equality holds when x ∈ {a 1 , a 2 }. So let x ∈ X ∖ {a 1 , a 2 } and, for a contradiction, suppose that H(a 1 , a 1 , x) = a 1 . By the associativity and quasitriviality of H, we then have

a 1 = H(a 1 , a 1 , x) = H(a 1 , H(a 1 , a 2 , a 2 ), x) = H(H(a 1 , a 1 , a 2 ), a 2 , x) = H(a 2 , a 2 , x) ∈ {a 2 , x},
which contradicts the fact that a 1 , a 2 and x are pairwise distinct. (b) Suppose to the contrary that a 1 is not a neutral element for H (the other case can be dealt with similarly). By Lemma 5(a) we have that H(a 1 , a 1 , y) = H(y, a 1 , a 1 ) = y for all y ∈ X. By assumption, there exists x ∈ X ∖ {a 1 , a 2 } such that H(a 1 , x, a 1 ) = a 1 . We have two cases to consider.

• If H(a 2 , x, a 2 ) = x, then by Lemma 5(a) we have that

H(x, a 2 , a 1 ) = H(H(x, a 1 , a 1 ), a 2 , a 1 ) = H(x, a 1 , H(a 1 , a 2 , a 1 )) = H(x, a 1 , a 2 ) = H(H(a 1 , a 1 , x), a 1 , a 2 ) = H(a 1 , H(a 1 , x, a 1 ), a 2 ) = H(a 1 , a 1 , a 2 ) = a 2 .
Also, by Lemma 5(a) we have that

x = H(x, a 1 , a 1 ) = H(H(a 2 , x, a 2 ), a 1 , a 1 ) = H(a 2 , H(x, a 2 , a 1 ), a 1 ) = H(a 2 , a 2 , a 1 ) = a 1 ,
which contradicts the fact that x ≠ a 1 . • If H(a 2 , x, a 2 ) = a 2 , then by Lemma 5(a) we have that

H(x, x, a 2 ) = H(x, H(a 2 , a 2 , x), a 2 ) = H(x, a 2 , H(a 2 , x, a 2 )) = H(x, a 2 , a 2 ) = x, and
H(a 1 , x, x) = H(a 1 , H(x, a 1 , a 1 ), x) = H(H(a 1 , x, a 1 ), a 1 , x) = H(a 1 , a 1 , x) = x.
By Lemma 5(a) we also have that

x = H(x, a 2 , a 2 ) = H(H(a 1 , x, x), a 2 , a 2 ) = H(a 1 , H(x, x, a 2 ), a 2 ) = H(a 1 , x, a 2 ) = H(a 1 , H(x, a 1 , a 1 ), a 2 ) = H(H(a 1 , x, a 1 ), a 1 , a 2 ) = H(a 1 , a 1 , a 2 ) = a 2 ,
which contradicts the fact that x ≠ a 2 .

◻

We now present some geometric considerations of quasitrivial operations. The preimage of an element x ∈ X under an operation F ∶ X n → X is denoted by

F -1 [x]. When X is finite, i.e. X = [k],
we also define the preimage sequence of F as the nondecreasing k-element sequence of the numbers

F -1 [x] , x ∈ [k]. We denote this sequence by F -1 .
Recall that the kernel of an operation

F ∶ [k] n → [k] is the equivalence relation Ker(F ) = {{x, y} F (x) = F (y)}. The contour plot of F ∶ [k] n → [k] is the undirected graph C F = ([k] n , E)
, where E is the non-reflexive part of Ker(F ), i.e., E = {{x, y} x ≠ y and F (x) = F (y)}. We say that two tuples

x, y ∈ [k] n are F -connected (or simply connected ) if {x, y} ∈ Ker(F ). Lemma 6. An operation F ∶ [k] n → [k] is quasitrivial if and only if it is idem- potent 3 and each (x 1 , ..., x n ) ∈ [k] n ∖ ∆ n [k] is connected to some (n ⋅ x) ∈ ∆ n [k] . Proof.
Clearly, F is quasitrivial if and only if it is idempotent and for any

(x 1 , ..., x n ) ∈ [k] n ∖ ∆ n [k] there exists i ∈ {1, ..., n} such that F (x 1 , ..., x n ) = x i = F (n ⋅ x i ). ◻
In the sequel we shall make use of the following two lemmas.

Lemma 7. For each x ∈ [k], the number of tuples (x 1 , . . . , x n ) ∈ [k] n with at least one component equal to x is given by k n -(k -1) n . Proof. Let x ∈ [k]. The set of tuples in [k] n with at least one component equal to x is the set [k] n ∖ ([k] ∖ {x}) n , and its cardinality is k n -(k -1) n since ([k] ∖ {x}) n ⊆ [k] n . ◻ Lemma 8. Let F ∶ [k] n → [k] be a quasitrivial operation. Then, for each x ∈ [k], we have F -1 [x] ≤ k n -(k -1) n . Proof. Let x ∈ [k]. Since F ∶ [k] n → [k] is quasitrivial, it follows from Lemma 6 that the point (n ⋅ x) is at most connected to all (x 1 , ..., x n ) ∈ [k]
n with at least one component equal to x. By Lemma 7, we conclude that there are exactly k n -(k -1) n such points. ◻

Recall that an element z ∈ X is said to be an annihilator for F if

F (x 1 , ..., x n ) = z, whenever (x 1 , ..., x n ) ∈ X n has at least one component equal to z.
Remark 1. A neutral element need not be unique when n ≥ 3 (e.g., F (x 1 , x 2 , x 3 ) ≡

x 1 + x 2 + x 3 (mod 2) on X = Z 2 )
. However, if an annihilator exists, then it is unique.

Proposition 9. Let F ∶ [k] n → [k] be a quasitrivial operation and let z ∈ [k].
Then z is an annihilator if and only if

F -1 [z] = k n -(k -1) n .
Proof. (Necessity) If z is an annihilator, then we know that

F (i⋅z, x i+1 , ..., x n ) = z for all i ∈ [n], all x i+1 , ..., x n ∈ [k] and all permutations of (i ⋅ z, x i+1 , ..., x n ). Thus, by Lemma 7 (n ⋅ z) is connected to k n -(k -1) n points. Finally, using Lemma 8 we get F -1 [z] = k n -(k -1) n . (Sufficiency) If F -1 [z] = k n -(k -1
) n , then by Lemmas 6 and 7 we have that

(n⋅z) is connected to the k n -(k-1) n points (x 1 , ..., x n ) ∈ [k] n containing at least one component equal to z. Thus, we have F (i ⋅ z, x i+1 , ..., x n ) = z for all i ∈ [n], all x i+1 , ..., x n ∈ [k]
and all permutations of (i⋅z, x i+1 , ..., x n ), which shows that z is an annihilator.

◻ Remark 2. By Proposition 9, if F ∶ [k] n → [k] is quasitrivial, then each ele- ment x such that F -1 [x] = k n -(k -1) n is unique.

Criteria for unique reductions and some enumeration results

In this section we show that an associative and quasitrivial operation F ∶ X n → X is uniquely reducible to an associative and quasitrivial binary operation if and only if F has at most one neutral element (Theorem 18). We also enumerate the class of associative and quasitrivial n-ary operations, which leads to a previously unknown sequence in the OEIS [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF] (Proposition 24). Let us first recall a useful result from [START_REF] Devillet | Characterizations of quasitrivial symmetric nondecreasing associative operations[END_REF].

Lemma 10 (see [START_REF] Devillet | Characterizations of quasitrivial symmetric nondecreasing associative operations[END_REF]Proposition 3.5]). Assume that the operation F ∶ X n → X is associative and reducible to associative binary operations G∶ X 2 → X and

G ′ ∶ X 2 → X. If G and G ′ are idempotent or have the same neutral element, then G = G ′ .
From Lemma 10, we immediately get a necessary and sufficient condition that guarantees unique reductions for associative operation that have a neutral element.

Corollary 11. Let F ∶ X n → X be an associative operation that is reducible to associative binary operations G∶ X 2 → X and G ′ ∶ X 2 → X that have neutral elements. Then, G = G ′ if and only if G and G ′ have the same neutral element.

Using Lemma 3, Corollary 11, and observing that (i) a binary associative operation has at most one neutral element, (ii) the neutral element of a binary reduction G∶ X 2 → X of an associative operation F ∶ X n → X is also a neutral element for F , and (iii) if e is a neutral element for an associative operation F ∶ X n → X and G∶ X 2 → X is a reduction of F , then G n-2 ((n -1) ⋅ e) (see Definition 2) is the neutral element for G, we can generalise Corollary 11 as follows.

Proposition 12. Let F ∶ X n → X be an associative operation, and let E F be the set of its neutral elements and R F of its binary reductions. If E F ≠ ∅, then for any G ∈ R F , there exists e ∈ E F such that G = G e . Moreover, the mapping σ∶ E F → R F defined by σ(e) = G e is a bijection. In particular, e is the unique neutral element for F if and only if G e is the unique binary reduction of F .

As we will see in Proposition 16, the size of E F , and thus of R F , is at most 2 whenever F is quasitrivial.

Let Q 2 1 (X) denote the class of associative and quasitrivial operations G∶ X 2 → X that have exactly one neutral element, and let A 2 1 (X) denote the class of associative operations G∶ X 2 → X that have exactly one neutral element e G ∈ X and that satisfy the following conditions:

• G(x, x) ∈ {e G , x} for all x ∈ X, • G(x, y) ∈ {x, y} for all (x, y) ∈ X 2 ∖ ∆ 2 X , • If there exists x ∈ X ∖ {e G } such that G(x, x) = e G , then x is unique and we have G(x, y) = G(y, x) = y for all y ∈ X ∖ {x, e G }. Note that Q 2 1 (X) = A 2 1 (X) = X X 2 when X = 1. Also, it is not difficult to see that Q 2 1 (X) ⊆ A 2 1 (X). Actually, we have that G ∈ Q 2 1 (X) if and only if G ∈ A 2 1 (X) and G -1 [e] = 1
, where e is the neutral element for G. A characterization of the class of associative and quasitrivial binary operations as well as Q 2 1 (X) can be found in [3, Theorem 2.1, Fact 2.4]. Recall that two groupoids (X, G) and (Y, G ′ ) are said to be isomorphic, and we denote it by (X,

G) ≃ (Y, G ′ ), if there exists a bijection φ∶ X → Y such that φ(G(x, y)) = G ′ (φ(x), φ(y)),
for every x, y ∈ X. The following straightforward proposition states, in particular, that any

G ∈ A 2 1 (X) ∖ Q 2 1 (X)
gives rise to a semigroup which has a unique 2-element subsemigroup isomorphic to the additive semigroup on Z 2 .

Proposition 13. Let G∶ X 2 → X be an operation. Then G ∈ A 2 1 (X) ∖ Q 2 1 (X) if and only if there exists a unique pair (x, y) ∈ X 2 ∖ ∆ 2
X such that the following conditions hold (a) ({x, y}, G {x,y} 2 ) ≃ (Z 2 , +), (b) G (X∖{x,y}) 2 is associative and quasitrivial, and (c) every z ∈ X ∖ {x, y} is an annihilator for G {x,y,z} 2 .

Proposition 14. Let F ∶ X n → X be an associative and quasitrivial operation.

Suppose that e ∈ X is a neutral element for F .

(a) If n is even, then F is reducible to an operation G ∈ Q 2 1 (X). (b) If n is odd, then F is reducible to the operation G e ∈ A 2 1 (X).
Proof. (a) By Theorem 1(a) we have that F is reducible to an associative and quasitrivial binary operation G∶ X 2 → X. Finally, we observe that G n-2 ((n -1) ⋅ e) is the neutral element for G.

(b) By Lemma 3 we have that F is reducible to an associative operation G e ∶ X 2 → X of the form [START_REF] Ackerman | A characterization of quasitrivial n-semigroups[END_REF] and that e is also a neutral element for G e . Since F is quasitrivial, it follows from (1) that G e (x, x) ∈ {x, e} for all x ∈ X. If X = 2, then the proof is complete. So suppose that X > 2 and let us show that G e (x, y) ∈ {x, y} for all (x, y) ∈ X 2 ∖ ∆ 2 X . Since e is a neutral element for G e , we have that G e (x, e) = G e (e, x) = x for all x ∈ X ∖ {e}. So suppose to the contrary that there are distinct x, y ∈ X ∖ {e} such that G e (x, y) ∈ {x, y}.

As G e is a reduction of F and F is quasitrivial, we must have G e (x, y) = e. But then, using the associativity of G e , we have that y = G e (e, y) = G e (G e (x, y), y) = G e (x, G e (y, y)) ∈ {G e (x, y), G e (x, e)} = {e, x}, which contradicts the fact that x, y and e are pairwise distinct. Now, suppose that there exists x ∈ X ∖ {e} such that G e (x, x) = e and let y ∈ X ∖ {x, e}. Since

y = G e (e, y) = G e (G e (x, x), y) = G e (x, G e (x, y)),
we must have G e (x, y) = y. Similarly, we can show that G e (y, x) = y.

To complete the proof, we only need to show that such an x is unique. Suppose to the contrary that there exists x ′ ∈ X ∖{x, e} such that G e (x ′ , x ′ ) = e. Since x, x ′ and e are pairwise distinct and

x ′ = G e (e, x ′ ) = G e (G e (x, x), x ′ ) = G e (x, G e (x, x ′ )),

and x = G e (x, e) = G e (x, G e (x ′ , x ′ )) = G e (G e (x, x ′ ), x ′ ),
we must have x = G e (x, x ′ ) = x ′ , which yields the desired contradiction. ◻

We observe that the associative operation F ∶ Z n 2 → Z 2 defined by

F (x 1 , . . . , x n ) ≡ n i=1 x i (mod 2), x 1 , . . . , x n ∈ Z 2 ,
has 2 neutral elements, namely 0 and 1, when n is odd. Moreover, it is quasitrivial if and only if n is odd. This also illustrates the fact that an associative and quasitrivial n-ary operation that has 2 neutral elements does not necessarily have a quasitrivial reduction. Indeed, when n is odd, G(x 1 , x 2 ) ≡

x 1 + x 2 (mod 2) and G ′ (x 1 , x 2 ) ≡ x 1 + x 2 + 1 (mod 2) on X = Z 2 are two distinct reductions of F but neither is quasitrivial. Clearly, if an associative operation F ∶ X n → X is reducible to an op- eration G ∈ Q 2 1 (X)
, then it is quasitrivial. The following proposition provides a necessary and sufficient condition for F to be quasitrivial when

G ∈ A 2 1 (X) ∖ Q 2 1 (X).
Proposition 15. Let F ∶ X n → X be an associative operation. Suppose that F is reducible to an operation G ∈ A 2 1 (X) ∖ Q 2 1 (X). Then, F is quasitrivial if and only if n is odd. Proof. To show that the condition is necessary, let x ∈ X ∖ {e} such that G(x, x) = e. If n is even, then

F (n ⋅ x) = G n 2 -1 ( n 2 ⋅ G(x, x)) = e, contradicting quasitriviality.
So let us prove that the condition is also sufficient. Note that G ∈ A 2 1 (X) ∖ Q 2 1 (X), and thus we only need to show that F is idempotent. Since F is reducible to G, we clearly have that F (n ⋅ x) = x for all x ∈ X such that G(x, x) = x.

Let y ∈ X ∖ {e} such that G(y, y) = e. Since n is odd, we have that

F (n ⋅ y) = G y, G n-1 2 -1 n -1 2 ⋅ G(y, y) = G(y, e) = y.
Hence, F is idempotent and the proof is now complete. ◻ It is not difficult to see that the operation F ∶ Z n n-1 → Z n-1 defined by

F (x 1 , . . . , x n ) ≡ n i=1 x i (mod (n -1)), x 1 , . . . , x n ∈ Z n-1 ,
is associative, idempotent, symmetric4 and has n -1 neutral elements. However, this number is much smaller for quasitrivial operations.

Proposition 16. Let F ∶ X n → X be an associative and quasitrivial operation. Proof. (a) By Theorem 1(a) we have that F is reducible to an associative and quasitrivial binary operation G∶ X 2 → X. Suppose that e 1 , e 2 ∈ X are two neutral elements for F . Since G is quasitrivial we have

e 2 = F ((n -1) ⋅ e 1 , e 2 ) = G(G n-2 ((n -1) ⋅ e 1 ), e 2 ) = G(e 1 , e 2 ) = G(e 1 , G n-2 ((n -1) ⋅ e 2 )) = F (e 1 , (n -1) ⋅ e 2 ) = e 1 .
Hence, F has at most one neutral element. (b) By Theorem 1(b) we have that F is reducible to an associative and quasitrivial ternary operation H∶ X 3 → X. For a contradiction, suppose that e 1 , e 2 , e 3 ∈ X are three distinct neutral elements for F . Since H is quasitrivial, it is not difficult to see that e 1 , e 2 , and e 3 are neutral elements for H. Also, by Proposition 14(b) we have that H is reducible to the operations G e1 , G e2 , G e3 ∈ A 2 1 (X). In particular, we have G e1 (e 2 , e 3 ) = G e1 (G e1 (e 1 , e 2 ), e 3 ) = H(e 1 , e 2 , e 3 ) = G e2 (G e2 (e 1 , e 2 ), e 3 ) = G e2 (e 1 , e 3 )

and

H(e 1 , e 2 , e 3 ) = G e3 (e 1 , G e3 (e 2 , e 3 )) = G e3 (e 1 , e 2 ).

Hence, H(e 1 , e 2 , e 3 ) ∈ {e 2 , e 3 } ∩ {e 1 , e 3 } ∩ {e 1 , e 2 }, which shows that e 1 , e 2 , e 3 are not pairwise distinct, and thus yielding the desired contradiction. ◻ Corollary 17. Let F ∶ X n → X be an operation and let e 1 and e 2 be distinct elements of X. Then F is associative, quasitrivial, and has exactly the two neutral elements e 1 and e 2 if and only if n is odd and F is reducible to exactly the two operations G e1 , G e2 ∈ A 2 1 (X) ∖ Q 2 1 (X). Proof. (Necessity) This follows from Propositions 12, 14, and 16 together with the observation that G e1 (e 2 , e 2 ) = e 1 and G e2 (e 1 , e 1 ) = e 2 .

(Sufficiency) This follows from Propositions 12 and 15. ◻

We can now state and prove the main result of this section.

Theorem 18. Let F ∶ X n → X be an associative and quasitrivial operation.

The following assertions are equivalent.

(i) Any binary reduction of F is idempotent.

(ii) Any binary reduction of F is quasitrivial.

(iii) F has at most one binary reduction.

(iv) F has at most one neutral element.

(v) F ((n -1) ⋅ x, y) = F (x, (n -1) ⋅ y) for any x, y ∈ X.

Proof. The implications (i) ⇒ ((ii) and (v)) and (v) ⇒ (iv) are straightforward. By Proposition 16 and Corollary 17 we also have the implications ((ii) or (iii)) ⇒ (iv). Hence, to complete the proof, it suffices to show that (iv) ⇒ ((i) and (iii)). First, we prove that (iv) ⇒ (i). We consider the two possible cases.

If F has a unique neutral element e, then by Proposition 12 G = G e is the unique reduction of F with neutral element e. For the sake of a contradiction, suppose that G is not idempotent. By Proposition 14 we then have that n is odd and

G ∈ A 2 1 (X) ∖ Q 2 1 (X). So let x ∈ X ∖ {e} such that G(x, x) ≠ x. Since G = G e , we must have G(x, x) = e. It is not difficult to see that F (y, (n -1) ⋅ x) = y = F ((n -1) ⋅ x, y) for all y ∈ X. Now, if there is i ∈ {2, . . . , n -1} such that F ((i -1) ⋅ x, e, (n -i) ⋅ x) = x,
then we have that i -1 and n -i are both even or both odd (since n is odd), and thus x = F ((i -1) ⋅ x, e, (n -i) ⋅ x) ∈ {G 2 (x, e, x), G 2 (e, e, e)} = {e}, which contradicts our assumption that x ≠ e. Hence, we have F ((i -1) ⋅ x, e, (n -i) ⋅ x) = e for all i ∈ {1, . . . , n}. Now, if X = 2, then the proof is complete since e and x are both neutral elements for F , which contradicts our assumption. So suppose that X > 2.

Since e is the unique neutral element for F , there exist y ∈ X ∖ {e, x} and i ∈ {2, . . . , n -1} such that

F ((i -1) ⋅ x, y, (n -i) ⋅ x) = x.
Again by the fact that n is odd, i -1 and n -i are both even or both odd, and thus x = F ((i -1) ⋅ x, y, (n -i) ⋅ x) ∈ {G 2 (x, y, x), G 2 (e, y, e)} = {G 2 (x, y, x), y}.

Since x ≠ y, we thus have that G 2 (x, y, x) = x. But then e,G(y,x)) = G(y, x) ∈ {x, y}, which contradicts our assumption that x, y, and e are pairwise distinct. Now, suppose that F has no neutral element and that G is a reduction of F that is not idempotent. So let x ∈ X such that G(x, x) ≠ x, and let y ∈ X ∖ {x, G(x, x)}. By the quasitriviality of F we have F ((n -1) ⋅ x, y) ∈ {x, y}. On the other hand, by the quasitriviality (and hence idempotency) of F and the associativity of G we have

e = G(x, x) = G(x, G 2 (x, y, x)) = G(G(x, x), G(y, x)) = G(
F ((n -1) ⋅ x, y) = F (F (n ⋅ x), (n -2) ⋅ x, y) = G(G n-2 (G n-1 (n ⋅ x), (n -2) ⋅ x), y) = G(G 2n-3 ((2n -2) ⋅ x), y) = G(G n-2 ((n -1) ⋅ G(x, x)), y) = F ((n -1) ⋅ G(x, x), y) ∈ {G(x, x), y}.
Since x, G(x, x), and y are pairwise distinct, it follows that F ((n-1)⋅x, y) = y, which implies that G(G n-2 ((n -1) ⋅ x), y) = y. Similarly, we can show that

G(y, G n-2 ((n -1) ⋅ x)) = y. Also, it is not difficult to see that G(G n-2 ((n -1) ⋅ x), G(x, x)) = G(x, x) = G(G(x, x), G n-2 ((n -1) ⋅ x)).
Furthermore, since F is idempotent and reducible to G, we also have that

G(G n-2 ((n -1) ⋅ x), x) = x = G(x, G n-2 ((n -1) ⋅ x)).
Thus G n-2 ((n -1) ⋅ x) is a neutral element for G and therefore a neutral element for F , which contradicts our assumption that F has no neutral element.

As both cases yield a contradiction, we conclude that G must be idempotent. The implication (iv) ⇒ (iii) is an immediate consequence of the implication (iv) ⇒ (i) together with Lemma 10. Thus, the proof of Theorem 18 is now complete. ◻ Remark 3. We observe that an alternative necessary and sufficient condition for the quasitriviality of a binary reduction of an n-ary quasitrivial semigroup has also been provided in [START_REF] Ackerman | A characterization of quasitrivial n-semigroups[END_REF]Corollary 3.16].

Theorem 18 together with Corollary 4 imply the following result.

Corollary 19. Let F ∶ X n → X be an operation. Then F is associative, quasitrivial, and has at most one neutral element if and only if it is reducible to an associative and quasitrivial operation G∶ X 2 → X. In this case, G is defined by G(x, y) = F (x, (n -1) ⋅ y).

Recall that a weak ordering on X is a binary relation ≲ on X that is total and transitive (see, e.g., [START_REF] Krantz | Foundations of measurement[END_REF] p. 14). We denote the symmetric part of ≲ by ∼. Also, a total ordering on X is a weak ordering on X that is antisymmetric. If (X, ≲) is a weakly ordered set, an element a ∈ X is said to be maximal for ≲ if x ≲ a for all x ∈ X. We denote the set of maximal elements of X for ≲ by M ≲ (X).

Given a weak ordering ≲ on X, the n-ary maximum operation on X for ≲ is the partial symmetric n-ary operation max n ≲ defined on

X n ∖ {(x 1 , . . . , x n ) ∈ X n ∶ M ≲ ({x 1 , . . . , x n }) ≥ 2} by max n ≲ (x 1 , . . . , x n ) = x i where i ∈ [n] is such that x j ≲ x i for all j ∈ [n].
If ≲ reduces to a total ordering, then clearly the operation max n ≲ is defined everywhere on X n . Also, the projection operations π 1 ∶ X n → X and π n ∶ X n → X are respectively defined by π 1 (x 1 , . . . , x n ) = x 1 and π n (x 1 , . . . , x n ) = x n for all x 1 , . . . , x n ∈ X.

Corollary 19 together with [9, Theorem 1] and [3, Corollary 2.3] imply the following characterization of the class of quasitrivial n-ary semigroups with at most one neutral element.

Theorem 20. Let F ∶ X n → X be an operation. Then F is associative, quasitrivial, and has at most one neutral element if and only if there exists a weak ordering ≲ on X and a binary reduction G∶ X 2 → X of F such that

G A×B = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ π 1 A×B or π 2 A×B , if A = B, max 2 ≲ A×B , otherwise, ∀A, B ∈ X ∼. (2)
Moreover, when X = [k], then the weak ordering ≲ is uniquely defined as follows:

x ≲ y ⇔ G -1 [x] ≤ G -1 [y] , x, y ∈ [k]. (3)
Now, let us illustrate Theorem 20 for binary operations by means of their contour plots. We can always represent the contour plot of any operation G∶ [k] 2 → [k] by fixing a total ordering on [k]. In Figure 1 (left), we represent the contour plot of an operation G∶ X 2 → X using the usual total ordering ≤ on X = {1, 2, 3, 4} 5 . It is not difficult to see that G is quasitrivial. To check whether G is associative, by Theorem 20, it suffices to show that G is of the form (2) where the weak ordering ≲ is defined on X by (3). In Figure 1 (right) we represent the contour plot of G using the weak ordering ≲ on X defined by [START_REF] Couceiro | Quasitrivial semigroups: characterizations and enumerations[END_REF]. We observe that G is of the form (2) for ≲ and thus by Theorem 20 it is associative.

Let ≤ be a total ordering on X.

An operation F ∶ X n → X is said to be ≤-preserving if F (x 1 , . . . , x n ) ≤ F (x ′ 1 , . . . , x ′ n ), whenever x i ≤ x ′ i for all i ∈ [n]
. Some associative binary operations G∶ X 2 → X are ≤-preserving for any total ordering on X (e.g., G(x, y) = x for all x, y ∈ X). However, there is no total ordering ≤ on X for which an operation

G ∈ A 2 1 (X) ∖ Q 2 1 (X) is ≤-preserving.
A typical example is the binary addition modulo 2.

Proposition 21. Suppose X ≥ 2. If G ∈ A 2 1 (X) ∖ Q 2 1 (X)
, then there is no total ordering ≤ on X that is preserved by G.

Proof. Let e ∈ X be the neutral element for G and let x ∈ X ∖ {e} such that G(x, x) = e. Suppose to the contrary that there exists a total ordering ≤ on X such that G is ≤-preserving. If x < e, then e = G(x, x) ≤ G(x, e) = x, which contradicts our assumption. The case x > e yields a similar contradiction. ◻ Remark 4. It is not difficult to see that any ≤-preserving operation F ∶ X n → X has at most one neutral element. Therefore, by Corollary 4 and Theorem 18 we conclude that any associative, quasitrivial, and ≤-preserving operation F ∶ X n → X is reducible to an associative, quasitrivial, and ≤-preserving operation G∶ X 2 → X. For a characterization of the class of associative, quasitrivial, and ≤-preserving operations G∶ X 2 → X, see [START_REF] Couceiro | Quasitrivial semigroups: characterizations and enumerations[END_REF]Theorem 4.5].

We now provide several enumeration results that give the sizes of the classes of associative and quasitrivial operations that were considered above when X = [k]. Recall that for any integers 0 ≤ ≤ k, the Stirling number of the second kind k is defined by

k = 1 ! i=0 (-1) -i i i k .
For any integer k ≥ 0, let q 2 (k) (resp. q n (k)) denote the number of associative and quasitrivial binary (resp. n-ary) operations on [k]. For any integer k ≥ 1, we denote by q 2 1 (k) the cardinality of Q 2 1 ([k]). Also, we denote by a 2 1 (k) the cardinality of A 2 1 ([k]). By definition, we have a 2 1 (1) = 1. In [START_REF] Couceiro | Quasitrivial semigroups: characterizations and enumerations[END_REF] the authors solved several enumeration problems concerning associative and quasitrivial binary operations. In particular, they computed q 2 (k) (see [START_REF] Couceiro | Quasitrivial semigroups: characterizations and enumerations[END_REF]Theorem 4.1]) as well as q 2 1 (k) (see [START_REF] Couceiro | Quasitrivial semigroups: characterizations and enumerations[END_REF]Proposition 4.2]). These sequences were also introduced in the OEIS [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF] as A292932(k) and A292933(k). The following result summarizes [START_REF] Couceiro | Quasitrivial semigroups: characterizations and enumerations[END_REF]Theorem 4.1] and [START_REF] Couceiro | Quasitrivial semigroups: characterizations and enumerations[END_REF]Proposition 4.2]. 1. First few values of q 2 (k), q 2 1 (k), q n 0 (k), q n 2 (k), q n (k) and a 2 1 (k)

k q 2 (k) q 2 1 (k) q n 0 (k) q n 2 (k) q n (k) a 2 1 (k) 1 1 1 0 0 1 1 2 4

Symmetric operations

In this section we refine our previous results to the subclass of associative and quasitrivial operations that are symmetric, and present further enumeration results accordingly.

We first recall and establish some auxiliary results.

Fact 25. Suppose that F ∶ X n → X is associative and surjective. If it is reducible to an associative operation G∶ X 2 → X, then G is surjective.

Lemma 26 (see [START_REF] Devillet | Characterizations of quasitrivial symmetric nondecreasing associative operations[END_REF]Lemma 3.6]). Suppose that F ∶ X n → X is associative, symmetric, and reducible to an associative and surjective operation G∶ X 2 → X. Then G is symmetric.

Proposition 27. If F ∶ X n → X is associative, quasitrivial, and symmetric, then it is reducible to an associative, surjective, and symmetric operation G∶ X 2 → X. Moreover, if X = [k], then F has a neutral element.

Proof. By Corollary 4, F is reducible to an associative operation G∶ X 2 → X. By Fact 25 and Lemma 26, it follows that G is surjective and symmetric.

For the moreover part, we only have two cases to consider.

• If G is quasitrivial, then by [START_REF] Couceiro | Quasitrivial semigroups: characterizations and enumerations[END_REF]Theorem 3.3] it follows that G has a neutral element, and thus F also has a neutral element. • If G is not quasitrivial, then by Proposition 16 and Theorem 18 F has in fact two neutral elements. ◻ Proposition 28 (see [START_REF] Ackerman | A characterization of quasitrivial n-semigroups[END_REF]Corollary 4.10]). An operation F ∶ X n → X is associative, quasitrivial, symmetric, and reducible to an associative and quasitrivial operation G∶ X 2 → X if and only if there exists a total ordering ⪯ on X such that F = max n ⪯ . Proposition 29. A quasitrivial operation F ∶ [k] n → [k] is associative, symmetric, and reducible to an associative and quasitrivial operation G∶

[k] 2 → [k] if and only if F -1 = (1, 2 n -1, . . . , k n -(k -1) n ).
Proof. (Necessity) Since G is quasitrivial, it is surjective and hence by Lemma 26 it is symmetric. Thus, by Proposition 28 there exists a total ordering ⪯ on X such that G(x, y) = max 2 ⪯ (x, y) for all x, y ∈ [k]. Hence F = max n ⪯ , which has an annihilator, and the proof of the necessity then follows by Proposition 9.

(Sufficiency) We proceed by induction on k. The result clearly holds for k = 1. Suppose that it holds for some k ≥ 1 and let us show that it still holds for k

+ 1. Assume that F ∶ [k + 1] n → [k + 1] is quasitrivial and that F -1 = (1, 2 n -1, . . . , (k + 1) n -k n ).
Let ⪯ be the total ordering on [k + 1] defined by

x ⪯ y if and only if F -1 (x) ≤ F -1 (y) , and let z = max k+1 ⪯ (1, . . . , k+1). Clearly,

F ′ = F ([k+1]∖{z}) n is quasitrivial and F ′-1 = (1, 2 n -1, . . . , k n -(k -1) n ).
By induction hypothesis we have that

F ′ = max n ⪯ ′ , where ⪯ ′ is the restriction of ⪯ to [k + 1] ∖ {z}. By Proposition 9, F -1 [z] = (k + 1) n -k n and thus F = max n ⪯ . ◻
We can now state and prove the main result of this section.

Theorem 30. Let F ∶ X n → X be an associative, quasitrivial, symmetric operation. The following assertions are equivalent.

(i) F is reducible to an associative and quasitrivial operation G∶ X 2 → X.

(ii) There exists a total ordering ⪯ on X such that F is ⪯-preserving.

(iii) There exists a total ordering ⪯ on X such that F = max n ⪯ . Moreover, when X = [k], each of the assertions (i) -(iii) is equivalent to each of the following assertions.

(iv) F has exactly one neutral element. (v) F -1 = (1, 2 n -1, . . . , k n -(k -1) n ).

Furthermore, the total ordering ⪯ considered in assertions (ii) and (iii) is uniquely defined as follows:

x ⪯ y if and only if

G -1 [x] ≤ G -1 [y] , x, y ∈ [k]. (4)
Moreover, there are k! operations satisfying any of the conditions (i) -(v).

Proof. (i) ⇒ (iii). This follows from Proposition 28. (iii) ⇒ (ii). Obvious. (ii) ⇒ (i). By Corollary 4 we have that F is reducible to an associative operation G∶ X 2 → X. Suppose to the contrary that G is not quasitrivial. From Theorem 18 and Proposition 16, it then follows that F has two neutral elements e 1 , e 2 ∈ X, which contradicts Remark 4.

(i) ⇔ (v). This follows from Proposition 29. (i) ⇒ (iv). This follows from Theorem 18 and Proposition 27. for all n × n matrices [c 1 ⋯ c n ] = [r 1 ⋯ r n ] T ∈ X n×n . In [6, Corollary 4.9] it was shown that associativity and bisymmetry are equivalent for operations F ∶ X n → X that are quasitrivial and symmetric. Thus, we can replace associativity with bisymmetry in Theorem 30.

Conclusion

In this paper we proved that any quasitrivial n-ary semigroup is reducible to a semigroup. Furthermore, we showed that a quasitrivial n-ary semigroup is reducible to a unique quasitrivial semigroup if and only if it has at most one neutral element. Moreover, we characterized the class of quasitrivial (and symmetric) n-ary semigroups that have at most one neutral element. Finally, when the underlying set is finite, this work led to four new integer sequences in the Sloane's OEIS [START_REF]The On-Line Encyclopedia of Integer Sequences[END_REF], namely, A308351, A308352, A308354, and A308362. Note however that there exist idempotent n-ary semigroups that are not reducible to a semigroup (for instance, consider the idempotent associative operation F ∶ R 3 → R defined by F (x, y, z) = x -y + z for all x, y, z ∈ R). This naturally asks for necessary and sufficient conditions under which an idempotent n-ary semigroup is reducible to a semigroup. This and other related questions constitute topics for future research.

  (a) If n is even, then F has at most one neutral element. (b) If n is odd, then F has at most two neutral elements.

Figure 1 .

 1 Figure 1. An associative and quasitrivial binary operation G on X = {1, 2, 3, 4} whose values on (1, 1), (2, 2), (3, 3) and (4, 4) are 1, 2, 3 and 4, respectively.

2 ( 2

 22 (iv) ⇒ (i). This follows from Lemma 3 and Theorem 18. The rest of the statement follows from[START_REF] Couceiro | Quasitrivial semigroups: characterizations and enumerations[END_REF] Theorem 3.3]. ◻ on [k]. Also, let qs n 2 (k) denote the number of associative, quasitrivial, and symmetric n-ary operations that have two neutral elements on [k]. Proposition 32. For each integer k ≥ 2, qs n 1(k) = qs 2 (k) = k!. Moreover, qs n 2 (k) = k! 2 ,and qs n (k) = 3k! 2 . Proof. By Theorems 30 and 18 and Lemma 26 we have that the number of associative, quasitrivial, and symmetric n-ary operations that have exactly one neutral element on [k] is exactly the number of associative, quasitrivial, and symmetric binary operations on[k]. By Theorems 18 and 30 this number is given by qs 2 (k) = k!. Also, by Corollary 17, Proposition 31, and Theorems 18 and 30, we have that q n and by Proposition 16 we have that q n (k) = qs n 1 (k) + qs n 2 (k) = 3k! 2 . ◻ Remark 5.Recall that an operation F ∶ X n → X is said to be bisymmetric if F (F (r 1 ), . . . , F (r n )) = F (F (c 1 ), . . . , F (c n ))

I.e., a set X endowed with an associative n-ary operation.

Quasitrivial operations are also called conservative operations[START_REF] Pouzet | A projection property[END_REF]. This property has been extensively used in the classification of constraint satisfaction problems into complexity classes (see, e.g,[START_REF] Bulatov | Conservative constraint satisfaction re-revisited[END_REF] and the references therein).

An operation F ∶ X n → X is idempotent if F (n ⋅ x) = x for all x ∈ X.

An operation F ∶ X n → X is symmetric if F (x 1 , . . . ,xn) is invariant under any permutation of x 1 , . . . , xn.

To simplify the representation of the connected components, we omit edges that can be obtained by transitivity.

Acknowledgements

Both authors would like to thank Jean-Luc Marichal and the anonymous referee for their useful comments and insightful remarks that helped improving the current paper.

The second author is supported by the Luxembourg National Research Fund under the project PRIDE 15/10949314/GSM.

Proposition 22. For any integer k ≥ 0, we have the closed-form expression

where q 2 (0) = q 2 (1) = 1. Moreover, for any integer k ≥ 1, we have q 2 1 (k) = k q 2 (k -1).

Proposition 23. For any integer k ≥ 2, we have a 2 1 (k) = kq 2 (k -1) + k(k -1)q 2 (k -2).

Proof. We already have that

). Now, let us show how to construct an operation

), so we have q 2 1 (k-1) possible choices to construct this restriction. This shows that a 2 1 (k) = q 2 1 (k)+kq 2 1 (k -1). Finally, by Proposition 22 we conclude that a

For any integer k ≥ 1 let q n 1 (k) (resp. q n 0 (k)) denote the number of associative and quasitrivial n-ary operations that have exactly one neutral element (resp. that have no neutral element) on [k]. Also, for any integer k ≥ 1, let q n 2 (k) denote the number of associative and quasitrivial n-ary operations that have two neutral elements on [k]. Clearly, q n (1) = q n 1 (1) = 1 and q n 2 (1) = 0. The following proposition provides explicit forms of the latter sequences. Table 1 below provides the first few values of all the previously considered sequences 6 .

Proposition 24. For any integer k ≥ 1 we have q n 1 (k) = q 2 1 (k) and q n 0 (k) = q 2 (k) -q 2 1 (k). Also, for any integer k ≥ 2 we have

By Theorem 18 we have that the number of associative and quasitrivial n-ary operations that have exactly one neutral element (resp. that have no neutral element) on [k] is exactly the number of associative and quasitrivial binary operation on [k] that have a neutral element (resp. that have no neutral element). This number is given by q 2 1 (k) (resp. q 2 (k) -q 2 1 (k)). Also, if n is even, then by Theorem 1(a) and Proposition 16(a) we conclude that q n (k) = q 2 (k) and q n 2 (k) = 0. Now, suppose that n is odd. By Corollary 17 and Propositions 22 and 23 we have that

. Finally, by Proposition 16 we have that q n (k) = q n 0 (k) + q n 1 (k) + q n 2 (k) = q 2 (k) + k 2 q 2 (k -2). ◻ 6 In view of Corollary 17, we only consider the case where n is odd for q n 2 (k) and q n (k).

Now, let us illustrate Theorem 30 for binary operations by means of their contour plots. In Figure 2 (left), we represent the contour plot of an operation G∶ X 2 → X using the usual total ordering ≤ on X = {1, 2, 3, 4}. In Figure 2 (right) we represent the contour plot of G using the total ordering ⪯ on X defined by (4). We then observe that G = max 2 ⪯ , which shows by Theorem 30 that G is associative, quasitrivial, and symmetric. Based on this example, we illustrate a simple test to check whether an operation F ∶ [k] n → [k] is associative, quasitrivial, symmetric, and has exactly one neutral element. First, construct the unique weak ordering ≾ on [k] from the preimage sequence F -1 , i.e., x ≾ y if

. Then, check if ≾ is a total ordering and if F is the maximum operation for ≾.

We denote the class of associative, quasitrivial, symmetric operations G∶ X 2 → X that have a neutral element e ∈ X by QS 2 1 (X). Also, we denote by AS 2 1 (X) the class of symmetric operations G∶ X 2 → X that belong to A 2 1 (X).

, where e is the neutral element for G. For each integer k ≥ 2, let qs n (k) denote the number of associative, quasitrivial, and symmetric n-ary operations on [k]. Also, denote by as 2 1 (k) the size of AS 2 1 ([k]). From Theorems 18 and 30 it follows that qs

Also, it is easy to check that as 2 1 (2) = 4. The remaining terms of the sequence are given in the following proposition.

Proposition 31. For every integer k ≥ 3, as


Éric Cousin
email: eric.cousin@enst-bretagne.fr

Gérald Ouvradou
email: gerald.ouvradou@enst-bretagne.fr

Pascal Pucci
email: pascal.pucci@enst-bretagne.fr

Samuel Tardieu
email: samuel.tardieu@enst.fr

PicoLibre: a free collaborative platform to improve students' skills in software engineering

Keywords: Skill development, new information technology tools, project activity, active learning, monitoring, sofware engineering, collaborative work, free software

The work described in this paper consists in the development of a pedagogical collaborative platform to improve the know-how of our students in software engineering and collaborative work. Its use in a free software context leads to greater commitment of the students and better reusability of their work.

I. INTRODUCTION

W HILE widely used in graduate or post-graduate French scientific cursus, project activities often miss some fundamental points. In particular, technical goals tend to prevail too heavily on organizational concerns. That is how, for instance, Computer Science students are trained to build software from scratch by themselves rather than to reuse existing code and to produce reusable code collaboratively. Yet, reusability has proven to be the most rational approach: it saves time and leads to more reliable software. Free Software practice relies heavily on this approach and that is where it takes place in our project.

In our view, opening projects to the Free Software community and using adequate collaborative tools can correct the forementioned bias. Work described in this paper consists in the development of a pedagogical collaborative platform, called PicoLibre. Its integration within our cursus helps to improve students' know-how in collaborative software engineering.

In the next section, the general context and the pedagogical needs are given. Third section deals with PicoLibre: a brief history of the project is made, and the main features of the platform are described. The fourth section explains how the use of the platform is promoted in our cursus, and makes some statements about the pedagogical impact. Crucial role of free software is underlined and some perspectives are drawn in the final conclusion.

II. CONTEXT AND PEDAGOGICAL GOALS

The GROUPE DES ÉCOLES DES TÉLÉCOMMUNICA-TIONS is composed of three main engineering schools and research centers. Highly-skilled students are selected by competitive examination and will be awarded a master's degree after three years of intensive study. We also have some hundred new PhD students every year.

During their cursus, all students have to complete several team projects in various fields (computer science, electronics, physics, ...). Each teacher will favour projects close to his research interests while students wish to achieve something they will be proud of.

According to French law, any software created by the students belongs to them, even if they have used college computing resources. To be able to use what was developed under our supervision for our research needs, we can request students to make a version of their work available under a Free Software license. Such a license grants the recipients (supervisors) the following freedoms [START_REF]The Free Software definition[END_REF]: "the freedom to run the program, for any purpose; the freedom to study how the program works and adapt it to one's needs; the freedom to redistribute copies; the freedom to improve the program and release one's improvements to the public, to the benefit of the whole community."

While using such a Free Software license preserves students' rights over their software (including the right to make a proprietary closed-source version if they want to), it gives supervisors the permission to use, enhance and distribute project results. Of course, unless the license allows it specifically, if a redistribution of the modified versions is made, it has to use the same license as the original, granting the recipients the same rights as given in the first place. The GNU General Public License (GPL) is one of those Free Software licenses. Any work derived from a software protected by the GPL cannot be released under a different license.

Our students are taught software engineering techniques. Typically, they have to accomplish certain pre-defined milestones in their project and defend their choices (planning, problem analysis, solution architecture, testing and conformance checking). In the best cases, they are required to use configuration management tools such as CVS1 in order to maximize team work efficiency and change tracking through the use of carefully thought-out commit messages.

However, this leads each group of students to use its own private repository2 ; at the end of the project, the supervisor only receives the final version, and all the history information which has been conscientiously collected and used throughout the project development is then lost. Most of the work done in a project is therefore never reused nor even integrated in a larger framework. Projects tend to terminate near the end of the semester, when the teacher is busy correcting exam papers and evaluating project results. Moreover, the missing history information makes it virtually impossible to have another team of students start another project on top of the existing one. For the same reasons, starting development of a larger Free Software project based on these grounds is difficult.

In fact, what we need is a system which supports both active project development and project memory facilities. We want not only to provide project development hosting, but also mailing-list features so that a new member can reconstitute the whole project evolution over time. Our view is that documentation and WWW pages associated with the project deserve the same treatment as the code itself, both during active development and between active phases. All these features constitute the aim of the PicoLibre platform which will now be described.

III. THE PICOLIBRE PROJECT

The first step of the project involved the technical specification of the platform. To this end, we studied existing hosting platforms3 looking how they could match our specific needs. Beyond this, our goal was also to look at the different technologies commonly used in such platforms to underpin our reflection.

We stated that our platform ought to have four major features. It should be:

• simple to use; • simple to install; • simple to manage;

• available in its entirety under the GPL.

It should be "simple to use" because, for a new user, the investment involved in learning platform usage has to be as light as possible compared to the code development load. Even for projects of moderate size, we want to promote the use of the platform, thus the induced overhead must remain at a reasonable level.

It should also be "simple to install" and "simple to manage": we wish to exchange platform experiments with other engineering schools and universities with the aim of improving our tool as well as our pedagogical methodology. Thus, the platform has to be suitable for adoption by light structures without heavy computer support.

The GPL license indeed makes it easier for anybody to use the platform, while it guarantees its durability: the software will never be closed, thus anybody may benefit from successive improvements.

Keeping this in mind, we mainly examined "Sourceforge" [START_REF]The SourceForge hosting platform[END_REF], the most famous Free Software hosting platform. Today, Sourceforge hosts as many as 30,000 open source 4 projects and 10 times more users. This platform is based on PHP technology 5 . Sourceforge seemed to us obviously an excellent groupware system for confirmed developers but, for our purpose, the training investment appeared far too great. In fact, Sourceforge offered too rich a framework for a novice and, moreover, it appeared also that it did not meet our two other criteria. Besides an extremely complex installation process, we noticed poor modularity of the software. This comes probably from the fact that the system had grown over time and was not, at the beginning, structured to become such a huge system. Trying to go back towards a simpler version of the platform does not provide a simple way to recover a simpler architecture.

Thus, the idea to try to adapt Sourceforge to our needs was forgotten. We therefore decided to look for another groupware kernel as the basis of our platform, which better matches our criteria.

From an architectural point of view, such a system is made up of two different parts: the frontend and the backend. The former has to support all user interactions typically through a web-interface. The latter is in charge of the project and user management. Basically, a platform hosts a set of projects. With each of them is associated a group of users. Each user has specific access rights to the various components of the project which he can manipulate thanks to dedicated tools. Technically, the backend implementation mainly relies on a database management system.

After analysing several groupware systems, we decided to adopt the PHP-GroupWare framework [START_REF]The PHP-GroupWare WWW site[END_REF] as the basis of our platform frontend. PHP-GroupWare (PH-PGW) is a free multi-user web-based groupware suite written in PHP. It also provides an API for developing additional applications. The greatest benefit we got with PHPGW was the high level of abstraction it offers for programming, thus speeding up and making our work more reliable. However, as PHPGW was based on a "per user" paradigm instead of the required "per project/per user", we had a significant amount of work to integrate this new paradigm into PHPGW. This will lead to a patch to be produced as our contribution feedback to this Free Software project.

In a cooperative and concurrent software development process, one of the most critical functions is the code versioning management. In our platform, this backend function is supported by CVS which is known to be a very secure system. But, this is insufficient if user access to the code repository is not strictly controlled. To this end, we adopted a Free Software produced by IDE-ALX which offers secure (i.e. crypted) web access to the CVS repository [START_REF]A chrooted SSH CVS server[END_REF]. As we used LDAP [START_REF] Tardieu | An open source implementation of the Lightweight Directory Access Protocol[END_REF] to register users on our platform, we had to adapt the IDEALX product. This add-on has been recently submitted as a patch to the project development group.

The first production version of our hosting platform PicoLibre was made available at the end of 2001. Here are the main features offered to a user registered in a hosted project group:

• Code and WWW repositories are under control of CVS and are accessed through web secured connection. Thanks to CVS, the history of these two data spaces is continuously traced, allowing them to be restored to any previous state. The WWW repository is used to feed the project web site. This site is activated as soon as the project is accommodated on the platform.

• Two mailing lists are automatically set up when a new project is hosted. Typically, one is used solely for developer communication and the other for both user and developer communication, but the usage remains up to the project group. Each group member may himself subscribe to or unsubscribe from the lists. These are continuously archived, thus the history of mailing group activity is permanently available for all members of the group. The mailing-list system manager is the Free Software Sympa [START_REF]The mailing-list manager Sympa[END_REF].

• A bug tracking tool allows the complete cycle life of a bug -once detected -to be traced. A bug cycle life begins with a bug submission which can be made by anyone. The submission is then analysed by a member of the project developer team. If it is just a misunderstanding and not a real bug, the submission is rejected and may lead to a new item in a FAQ (Frequently Asked Questions). Conversely, a "real" bug is stated as open. When a developer deals with it, the bug becomes assigned. Then, the developer submits a patch intended to fix the bug. When the patch has been successfully tested, the bug is marked closed and is archived. The patch is then integrated into the so called "project development version"(as opposed to the "stable version").

• A task scheduler board allows a set of tasks concerning the project development to be scheduled and human and material resources to be allocated to them. For instance, a bug fixing task may be planned with this tool.

• Project documentation and downloading tools are also provided. Various web pages are automatically set up at the project's creation. To do so, questionnaries are submitted to the project creator. As soon as a first version of the code is available, it becomes accessible to authorized people (see below) through a downloading tool.

The administrator of the platform (user and project management) is offered several tools. For instance, visibility of a project is controlled via a private or public status. A private project involves a restricted access limited to a set of identified users. Conversely, all components of a public project are accessible to any registered user. There is also an anonymous access mode which allows to access to certain parts of a public project (e.g. the code downloading) without requiring to be registered on the platform. In a project user group, two categories of user are implemented: administrators and developers. Administrators may set up and modify at any time the access rights of other group's users.

Other functionnalities are currently under development. For instance: a documentation formatter based on XML, a search engine to browse projects hosted on the platform, a project activity ranking tool, a platform monitoring system, and so on. Of course, an important point to keep in mind is the pedagogical objective of PicoLibre. Each new tool considered has to be examined from this point of view (i.e. real need, real usefulness, impact on the user-interface, time needed to master this new functionality, etc.). To this end, PicoLibre architecture is modular and flexible, as it also allows easy customisation of each site installed. Soon, a Debian package of PicoLibre will be available and so, the installation process onto corresponding platforms will be dramatically improved. In the future, we intend to produce software packages for the most common Unix-GNU/linux based platforms.

IV. INTEGRATING PICOLIBRE IN OUR CURSUS : FIRST RESULTS

We have been using our platform for several months now [START_REF]PicoLibre platform hosted by ENST Bretagne[END_REF]. Users of the first prototype -whose name was Serveur Libre -were mainly the handful of students that were developing it. Nevertheless, this brought us enough feedback to confirm the previously identified needs and to design the new version of the platform, PicoLibre. This was soon judged stable and mature enough to enter operational use. Introducing PicoLibre into our cursus was therefore possible.

To remain in the Free Software spirit, we wanted to keep the same approach as the one followed since the beginning of the project. That is, students who take part in the development of the platform itself, or simply use it to host their project, should do so on a voluntary basis.

The first pedagogical challenge is therefore to encourage them to use the platform. Given that some software engineering courses are already taught in our cursus, only a short additional introduction to our platform is therefore required ; the main functionalities are presented in one class, and then practiced directly on the platform with some demonstration projects. As the overall use of the platform is simple, the keypoint lies in understanding the use of CVS; this is rather simple. Students then know enough about the platform to make their choice accordingly.

Taking into account observations based on the previous prototype and first feedbacks with PicoLibre, here are some findings and what we have learned about the pedagogical impact of the use of this platform:

• Good practice advice in software engineering is too often considered as obvious and theoretical by students. Programming on a small scale, as it is mostly the case in their cursus, generally does not allow them to realize how difficult collaborative development may be, and that some discipline should be followed. The use of the platform makes them have another outlook on these aspects and improves their practices.

• When hosted on the platform, students'work has a better chance to be really used. As an example, our project itself involved more than fifteen student projects that the platform allowed us to proprerly supervise and integrate.

• As we already stated in a previous article [START_REF] Ouvradou | Valorisation de projets d'étudiants sous forme de logiciel libre[END_REF], opening their projects to the Free Software community has a great impact on the way students work. Knowing that everybody can access their work 6 increases their overall motivation and their commitment to produce reusable code.

• As far as evaluation of students' work is concerned, we take for granted that, most of the time, the students' code is not inspected by supervisors. Evaluation is therefore mainly based on the overall aspect of the software, and accompanying reports. From this point of view, the platform allows supervisors to follow more closely what is going on, but does not help very much. Furthermore, the underlying philosophy of PicoLibre is rather to encourage students to reuse existing code and to work together with other students. This raises a new difficulty. Part of this concern may probably be tackled with some new monitoring facilities to be added soon to the platform.

Given the mentioned pedagogical interests, a midterm objective is therefore that each student should use the platform for at least one of his projects so as to fully understand the usefulness of the different tools offered, and to acquire know-how in collaborative engineering.

V. CONCLUSION AND PERSPECTIVES

We were surprisingly pleased by the success of our experiments. While we were quite anxious not to hamper students' project development because of bugs or inadequacies in our platform, we did obtain positive feedback and suggestions. The use of PicoLibre has shown good pedagogical advantages.

It is to be noted that Free Software plays a central role in our approach. We have just mentioned how sharing their code with the Free Software community may positively influence students' work. Another point is that development of the PicoLibre platform itself would not have been possible without the availability of all those software components described in section 3. Finally, Pi-coLibre is also a Free Software 7 . As it is self-hosting [START_REF]PicoLibre platform hosted by ENST Bretagne[END_REF], anyone with internet access can get the latest version and install it on his computer to host public or private software development, WWW sites, internal documents, and so on. Anyone can also contribute by adding features, fixing bugs or writing documentation. This allows any other university to try the platform, and should favour the growth of the project.

The first publicly available release has been tested on Debian GNU/Linux systems. We are now extending the range of supported platforms; the first one on the list will be FreeBSD, another high-performance Free Software Unix system. Our goal is to run PicoLibre on any operating system where PHP Groupware (the web software framework we use), CVS, SSH and Sympa are supported; that includes virtually any Unix-like system.

Monitoring facilities are being added to the PicoLibre platform. These facilities will help a supervisor to analyse what is going on in a project and within a team. For example, the number of changes made by each team member will be available, as will be the number of features added. Also, the vitality of the project will be evaluated; this should ease early detection of blocking problems.

We have also been working with numerous partners on the CoopX [START_REF]The CoopX project[END_REF] project to define a communication protocol allowing integrated development sites such as Pi-coLibre to exchange information [START_REF] Wazni | Protocoles d'échange entre serveurs coopératifs[END_REF]. For example, a developer registered on a PicoLibre platform could use any GNU Savannah site without registering again. Another concern is project migration: many people want to be able to move from one platform to another if they are not satisfied with the service quality or if they need

CVS (Concurrent Versions System) is the most commonly used version management tool in the Free Software world.

The repository is the place where files are stored permanently, along with their full history.

A hosting platform offers a set of computer resources in order to store all the components of a project (including user accounts), and tools for facilitating cooperative work.

"Open Source" is a less restrictive definition than "Free Software", see http://www.gnu.org/philosophy/free-sw.html

PHP is a widely-used general-purpose scripting language that is especially suited for Web development and can be embedded into HTML, see http://www.php.net/

Of course, only some of the projects meet all the requirements to really be valuable for the community.

Acknowledgments

The PicoLibre platform, which is itself hosted on a Pi-coLibre installation at ENST de Bretagne, demonstrates a successful cooperative work, integrating direct and indirect contributions from tenths of people. We want to thank them all, especially our collegues and students from the Groupe des Écoles des Télécommunications.

different functionalities. More information can be found on the WWW site of the project [START_REF]PicoLibre WWW site[END_REF].


Marine Baron
email: marine.baron@aphp.fr

Laurence Simon

Stephanie Poulain

Veronique Leblond

Stéphanie Poulain

Véronique Leblond

How Recent Advances in Biology of Waldenström's Macroglobulinemia May Affect Therapy Strategy

Keywords: Waldenström, Personalized treatment, CXCR4 mutation, MyD88, Next-generation sequencing

published or not. The documents may come

Introduction

Waldenström macroglobulinemia (WM) is a rare lymphoplasmacytic lymphoma characterized by cellular bone marrow (BM) involvement and monoclonal IgM production. In many cases, the diagnosis is fortuitous and there is no need for immediate therapeutic intervention. Criteria for treatment initiation are cytopenia (BM involvement, immunological destruction, or chronic inflammation), constitutional symptoms, hyperviscosity syndrome, and complications related to IgM activity (anti-myelin-associated glycoprotein (MAG) neurop-athy, cold agglutinins disease ...), or to deposits (AL amyloidosis ...) [START_REF] Simon | How we manage patients with Waldenström macroglobulinaemia[END_REF].

At the moment, therapeutic choices are guided by patients' characteristics (age, general condition, and comorbidities) and urgent efficacy need but not by biological disease's characteristics [START_REF] Leblond | Treatment recommendations from the eighth international workshop on Waldenström's macroglobulinemia[END_REF]. Most of the patients are initially treated with immunochemotherapies with rituximabbased regimens.

Recently, various recurrent mutations, including MyD88 and CXCXR4 mutations, have been described with different clinical presentations, prognoses, and responses to treatment [START_REF] Hunter | The genomic landscape of Waldenstrom macroglobulinemia is characterized by highly recurring MYD88 and WHIM-like CXCR4 mutations, and small somatic deletions associated with B-cell lymphomagenesis[END_REF][START_REF] Treon | MYD88 L265P somatic mutation in Waldenström's macroglobulinemia[END_REF]. These mutations could influence future therapeutic choices. Furthermore, the functional consequences of those mutations could be targeted and new therapies are on development. Finally, p53 defective pathways have been reported in various lymphoproliferative disorders, such as chronic lymphocytic leukemia (CLL), and are associated with poor prognosis and poor treatment responses [START_REF] Tessoulin | p53 dysregulation in B-cell malignancies: more than a single gene in the pathway to hell[END_REF]. Targeted therapies, such as Bruton tyrosine kinase (BTK) inhibitors, are used to bypass this p53 defects [START_REF] Farooqui | Ibrutinib for previously untreated and relapsed or refractory chronic lymphocytic leukaemia with TP53 aberrations: a phase 2, single-arm trial[END_REF]. In MW, p53 defective pathways and their functional consequences have been very recently reported.

This review highlights how recent advances in biology of MW could affect therapies.

Body Headings

MyD88 Mutation: a Diagnosis Tool and Prognostic Factor?

The L265P mutation of myeloid primary differentiation 88 (MyD88) has been first reported in more than 90% of MW by Treon 6 years ago using whole-genome sequencing and has been rapidly confirmed by other teams [START_REF] Treon | MYD88 L265P somatic mutation in Waldenström's macroglobulinemia[END_REF][START_REF] Poulain | MYD88 L265P mutation in Waldenstrom macroglobulinemia[END_REF]. MyD88 is a signaling adaptive protein that activates the NF-κB pathway following stimulation with Toll-like receptors (TLRs) and receptors for IL1 and IL18, as described in Fig. 1. Following ligand binding, receptors are associated with MyD88 as a homodimer. Then, the IL1 receptor kinase 4 (IRAK4) is recruited to MyD88 and phosphorylates IRAK1 and IRAK2 to form a complex known as the "myddosome" which activates the NF-κB pathway and the mitogen-activated protein kinase (MAPK) pathway. MyD88 activating mutation is responsible for enhancement cell survival through increase in NF-κB activity, JAK-STAT3 signaling, and consequently cytokine production. This mutation is not specific of WM and is found, to a lesser extent, in other indolent or aggressive lymphoproliferative disorders [START_REF] Treon | MYD88 L265P somatic mutation in Waldenström's macroglobulinemia[END_REF][START_REF] Xu | MYD88 L265P in Waldenström macroglobulinemia, immunoglobulin M monoclonal gammopathy, and other B-cell lymphoproliferative disorders using conventional and quantitative allele-specific polymerase chain reaction[END_REF]. Indeed, MYD88 L265P mutation is detected in nearly half of IgM monoclonal gammopathy of undetermined significance (MGUS) patients, 10% of splenic marginal zone lymphoma patients, 4% of CLL patients, one quart of activated B cell (ABC) diffuse B cell lymphoma patients (DLBCL), and 70% of central nervous system (CNS) DLBCL patients [START_REF] Xu | MYD88 L265P in Waldenström macroglobulinemia, immunoglobulin M monoclonal gammopathy, and other B-cell lymphoproliferative disorders using conventional and quantitative allele-specific polymerase chain reaction[END_REF][START_REF] Yu | Prognostic value and efficacy evaluation of novel drugs for cytogenetic aberrations in multiple myeloma: a meta-analysis[END_REF]. The mutation is not found in IgG MGUS and myeloma patients or healthy donors. A highest BM disease burden and a better prognosis were initially reported in MyD88-mutated (mut) versus MyD88 wild-type (WT) patients but these data are still controversial [START_REF] Abeykoon | MYD88 mutation status does not impact overall survival in Waldenström macroglobulinemia[END_REF][START_REF] Treon | Somatic mutations in MYD88 and CXCR4 are determinants of clinical presentation and overall survival in Waldenstrom macroglobulinemia[END_REF][START_REF] Treon | MYD88 wild-type Waldenstrom macroglobulinaemia: differential diagnosis, risk of histological transformation, and overall survival[END_REF]. Indeed, Treon showed, in 326 patients, an estimated 10-year survival of 73% versus 90%, in MyD88 WT versus MyD88 mut patients respectively [START_REF] Treon | MYD88 wild-type Waldenstrom macroglobulinaemia: differential diagnosis, risk of histological transformation, and overall survival[END_REF]. But, more recently, no difference in tumor burden or overall survival were found by Mayo Clinic group in 219 patients [START_REF] Abeykoon | MYD88 mutation status does not impact overall survival in Waldenström macroglobulinemia[END_REF]. This difference can be explained because patients with indolent disease were excluded and overall survival was estimated in symptomatic WM patients only. Indeed, time to progression from smoldering to active disease seemed to be shorter for MyD88 WT patients A constitutive activation of nuclear factor-κB (NF-κB) was demonstrated in WM cells. Genes of canonical and noncanonical NF-κB pathways, CXCR4, Toll-like receptors (TLR), and B cell receptor (BCR) may be activated deregulation of survival, apoptosis and proliferation, which may lead to WM clone expansion and tumor progression. Reprinted from Magierowicz et al. [START_REF] Magierowicz | Working toward a genomic prognostic classification of Waldenström macroglobulinemia: C-X-C chemokine receptor type 4 mutation and beyond[END_REF], with permission from Elsevier in this latest report (1.8 versus 2.9 years without reaching statistical significance). In summary, more prospective studies are needed to conclude.

MyD88 Mutation: Does It Influence Treatment Response?

Bruton tyrosine kinase (BTK) is a critical node in B cell receptor (BCR) signaling cascades, mediating the signal from BCR to downstream pathways such as NF-κB, phosphatydilinositol3 kinase (PI3K), AKT, and NFAT. In WM cells with MyD88 L265P, MyD88 is preferentially complexed to phosphorylated BTK (pBTK), yet little complexing is observed in lymphoma cells with WT Myd88 [START_REF] Yang | A mutat i o n i n M Y D 8 8 ( L 2 6 5 P ) s u p p o r t s t h e s u r v i v a l o f lymphoplasmacytic cells by activation of Bruton tyrosine kinase in Waldenström macroglobulinemia[END_REF]. The level of pBTK is higher in WM cells with MyD88 L265P than in WT Myd88 cells. Ibrutinib is a BTK inhibitor, now commonly used in lymphoproliferative disorders, irrespectively of MyD88 mutation. It reduces the binding strength of BTK to MyD88 L265P [START_REF] Yang | A mutat i o n i n M Y D 8 8 ( L 2 6 5 P ) s u p p o r t s t h e s u r v i v a l o f lymphoplasmacytic cells by activation of Bruton tyrosine kinase in Waldenström macroglobulinemia[END_REF]. The first report of ibrutinib-based treatment f in WM patients has demonstrated a great efficacy and suggested worst response rates in MyD88 WT patients but was assessed in only five MyD88 WT patients. Those patients had no major response to ibrutinib [15•, 16]. Secondarily, no conclusion have been drawn in the arm C of iNNOVATE study, as only one patient had MYD88 WT genotype [START_REF] Dimopoulos | Ibrutinib for patients with rituximab-refractory Waldenström's macroglobulinaemia (iNNOVATE): an open-label substudy of an international, multicentre, phase 3 trial[END_REF]. This study evaluated ibrutinib plus rituximab versus placebo plus rituximab for patients with relapsed MW. The MyD88 WT patient obtained a stable disease whereas overall response rate (ORR) was 90%. Recently, Dimopoulos reported a slight lower response rate in 11 MyD88 WT patients versus 58 mut patients treated with rituximab plus ibrunitib (81% of overall response rate versus 97%) [START_REF] Dimopoulos | Phase 3 trial of ibrutinib plus rituximab in Waldenström's macroglobulinemia[END_REF]. MyD88 mutational status did not seem to impact rituximab response rate in the rituximab plus placebo group. These minor differences in response rates with regard to the MyD88 mutational status did not affect the progression free survival (PFS). These results may be interpreted with caution in this small series. The question whether MyD88 mutation is a predictive factor in WM patients treated with ibrutinib is still open. Conventional immunochemotherapies should be preferred in case of MyD88 WT pending further information. Few data are available to compare therapeutic responses to standard immunochemotherapies according to mutational status. Paludo et al. reported that regimens with bendamustine and rituximab (BR) or with dexamethasone, rituximab, and cyclophosphamide (DRC) are equally effective in 38 L265P and 10 WT MyD88 patients with similar response rates and similar time to new treatment [START_REF] Paludo | Bendamustine and rituximab (BR) versus dexamethasone, rituximab, and cyclophosphamide (DRC) in patients with Waldenström macroglobulinemia[END_REF]. Very recently, FILO French group confirmed that MYD88 mutations did not impact on disease response for 69 patients treated with BR [START_REF] Laribi | Bendamustine plus rituximab in newly-diagnosed Waldenstrom macroglobulinemia patients. A study on behalf of the French Innovative Leukemia Organization (FILO)[END_REF]. We can conclude that MyD88 mutation do not impact on response to conventional chemotherapies.

The Myddosome: Functional Consequences and Targeted Strategies

New therapeutic approaches targeting MyD88-driven pathway, so-called myddosome (which includes IRAK1/4 and BTK among others), have been developed or are still on development, such as BTK, PI3K, TLR, IRAK, and hematopoietic cell kinase (HCK) inhibitors, as well as targeting the MyD88 molecule itself.

IRAK Inhibitors After stimulation, MyD88 forms complexes with IRAK4. MyD88 L265 P mutation activates downstream signaling pathway, such as IRAK1/4. Indeed, compared with the WT MyD88 protein, the mutated MyD88 protein has enhanced IRAK4 and pIRAK1 binding and can form the myddosome without external stimuli [START_REF] Yang | A mutat i o n i n M Y D 8 8 ( L 2 6 5 P ) s u p p o r t s t h e s u r v i v a l o f lymphoplasmacytic cells by activation of Bruton tyrosine kinase in Waldenström macroglobulinemia[END_REF][START_REF] Ngo | Oncogenically active MYD88 mutations in human lymphoma[END_REF]. Furthermore, IRAK1/4 signaling may contribute to persistent WM cell survival following ibrutinib treatment since WM patients cells following 6 months of continued ibrutinib treatment demonstrated highly active IRAK1 and IRAK4, but not BTK [22•]. Treatment of primary WM cells taken from untreated patients, patients on ibrutinib therapy, as well as MYD88-mutated WM cells lines with ibrutinib, and a toolbox IRAK4/IRAK1 inhibitor resulted in more robust reductions in NF-κB signaling, and at least additive tumor cell killing versus either agent alone [22•]. This study provides a framework for the development and investigation of IRAK inhibitors, alone and in combination with ibrutinib in WM patients.

HCK Inhibitors HCK is a member of the SRC family of protein tyrosine kinases, and one of the most aberrantly upregulated genes in WM cells [START_REF] Gutiérrez | Gene expression profiling of B lymphocytes and plasma cells from Waldenström's macroglobulinemia: comparison with expression patterns of the same cell counterparts from chronic lymphocytic leukemia, multiple myeloma and normal individuals[END_REF]. HCK is activated by interleukin-6 among others. Yang et al. investigated the role of HCK in WM, and the impact of MyD88 mutation and ibrutinib on the transcriptional regulation and activation of HCK [24•]. They demonstrated that HCK transcription is driven by mutated MyD88 and that HCK is a determinant of survival in MyD88-mutated cells, whereas HCK expression was virtually absent in WT cells. Ibrutinib is also a potent nonconvalent inhibitor of several SRC family members including HCK. Ibrutinib and a preclinical HCK inhibitor attenuated HCK activation and impact survival in Myd88 WM-mutated cells. HCK should represent a novel target for therapeutic development in MyD88-mutated cells.

MyD88 Inhibitors MyD88 dimerization seems to be necessary for assembly of the myddosome, which is composed of MyD88 dimers that recruit and activate IRAK1/2/4 complex and can then trigger NF-κB activation mediating cell growth and survival signaling. In contrast to native Myd88, mutated MyD88 protein can assemble without external stimuli and trigger constitutive NF-κB activation. First report of L265P MyD88 mutation in WM patients has shown marked decrease in nuclear staining of NF-κB p65, as well as decreased IkBα and NF-κB p65 phosphorylation after culture of MyD88-mutated cells with an inhibitor of Myd88 homodimerization [START_REF] Treon | MYD88 L265P somatic mutation in Waldenström's macroglobulinemia[END_REF]. In contrast, WT WM cells were unaffected by the inhibition of MyD88 signaling. More recently, preclinical studies have demonstrated that interference of myddosome assembly within selected regions of the Toll/interleukin-1 receptor (TIR) domain and death domain (DD) of MyD88 protein can impact growth and survival signaling of MyD88L265P-mutated WM cells [25•]. Interference was obtained with mini-peptides designed to compete with MyD88 TIR and DD domain interactions. These findings provide a framework for developing this type of agents.

TLR Inhibitors The MyD88 mutation has been shown to overactivate TLR7 and 9 mediating signaling pathways. Pharmacological inhibitors of TLR9 have been considered and clinical trial has been proposed with an oligonucleotide inhibiting TLRs 7.8 and 9 (IMO-8400) in WM patients but results are not known yet. IMO-8400 has also been studied in mouse xenografts models, alone or in combination with rituximab. It was well tolerated and exerted potent anti-tumor activity against B cell lymphomas with the MYD88 L265P oncogenic mutation, providing a strong rationale to evaluate this treatment for patients harboring MyD88 mutation [START_REF] Wang | Novel approach to the potential treatment of patients with B-cell lymphomas harboring the MYD88 L265P mutation: combination treatment with TLR antagonist and rituximab[END_REF]. Nevertheless, and unexpectedly, Wang et al. have then reported that interference with TLR9 signaling paradoxically promotes accumulation of MyD88-mutated lymphoplasmablasts in vivo [START_REF] Wang | Inhibiting TLR9 and other UNC93B1-dependent TLRs paradoxically increases accumulation of MYD88L265P plasmablasts in vivo[END_REF]. These conclusions raise the possibility that inhibiting these receptors may be associated with adverse implications and have to be specified.

TCR-Based Immunotherapy Moreover, MyD88 L265P containing peptides can elicit human leukocyte antigen (HLA) class-1 restricted cytotoxic T cell response, supporting the potential for T cell receptor (TCR)-based immunotherapy. Indeed, the immune system can recognize, and to some extent, eradicate tumor cells but this anti-tumor response is often inefficient. Antigen-specific immunotherapy holds the potential to induce and boost clinically effective cancer T cell responses. Tumor-specific neoepitopes can derive from protein-alteration mutational events. Nelde et al. have identified a set of HLA class 1 MyD88 L265P-derived ligands that elicit specific T cell responses [START_REF] Nelde | HLA class I-restricted MYD88 L265P-derived peptides as specific targets for lymphoma immunotherapy[END_REF]. These data highlight the potential of MyD88 L265P-mutation-specific peptides-based immunotherapy as a novel personalized treatment approach.

CXCR4 Mutation: a Prognostic Factor?

CXCR4 is a chemokine receptor that promotes WM survival, migration, and adhesion to the BM stroma through interaction with its ligand CXCL12 (or SDF-1) [START_REF] Ngo | SDF-1/ CXCR4 and VLA-4 interaction regulates homing in Waldenstrom macroglobulinemia[END_REF]. WM cells express CXCR4. Activating C-X-C chemokine receptor type-4 (CXCR4) frameshift or nonsense mutations in the C terminal tail was demonstrated in approximately 30% of WM patients [START_REF] Hunter | The genomic landscape of Waldenstrom macroglobulinemia is characterized by highly recurring MYD88 and WHIM-like CXCR4 mutations, and small somatic deletions associated with B-cell lymphomagenesis[END_REF]. The location of these somatic mutations is similar to that observed in the germline of patients with warts, hypogammaglobulinemia, infections, and myelokathexis (WHIM) syndrome, a congenital immunodeficiency disorder characterized by chronic noncyclic neutropenia [START_REF] Hernandez | Mutations in the chemokine receptor gene CXCR4 are associated with WHIM syndrome, a combined immunodeficiency disease[END_REF]. Importantly, 98% CXCR4 mut patients are also MyD88 L265P positive, suggesting a potential cooperation between these pathways. These activating mutations impair receptor internalization and result in persistent CXCR4 activation and BM myeloid cell trafficking. At least, 40 different CXCR4 somatic mutations have been identified in patients with WM. Furthermore, genes related to CXCR4 signaling were upregulated (CXCR4, CXCL12/SDF-1, and VCAM1), regardless of the CXCR4 mutational status [START_REF] Hunter | Transcriptome sequencing reveals a profile that corresponds to genomic variants in Waldenström macroglobulinemia[END_REF]. This confirms the crucial role of CXCR4 pathway in WM pathophysiology. Preclinical studies with the most common CXCR4 S338X mutation in WM have shown sustained signaling of AKT, ERK, and BTK following CXCL12 binding in comparison with wildtype CXCR4, as well as increased cell growth and survival of WM cells. CXCR4 mutations are associated with complex genomic aberrations [START_REF] Magierowicz | Working toward a genomic prognostic classification of Waldenström macroglobulinemia: C-X-C chemokine receptor type 4 mutation and beyond[END_REF][START_REF] Poulain | Genomic landscape of CXCR4 mutations in Waldenström macroglobulinemia[END_REF]. Patients with CXCR4 mutations demonstrate lower adenopathies, delineating differences in disease tropism, and nonsense mutations are associated with higher IgM production and hyperviscosity [START_REF] Treon | Somatic mutations in MYD88 and CXCR4 are determinants of clinical presentation and overall survival in Waldenstrom macroglobulinemia[END_REF]. Overall survival is not adversely impacted by CXCR4 mutational status [START_REF] Treon | Somatic mutations in MYD88 and CXCR4 are determinants of clinical presentation and overall survival in Waldenstrom macroglobulinemia[END_REF][START_REF] Dimopoulos | Ibrutinib for patients with rituximab-refractory Waldenström's macroglobulinaemia (iNNOVATE): an open-label substudy of an international, multicentre, phase 3 trial[END_REF].

CXCR4 Mutation: Does It Influence Treatment Response?

A potential clinical impact of CXCR4 mutation has been suggested by several teams, by showing drug resistance in vitro [33•, 34]. Indeed, Rocarro and colleagues investigated the functional role of this variant. CXCR4-mutated mice presented with a significant dissemination of tumor cells to distant organs and a reduced survival. WMmutated cells demonstrated resistance to BTK as well as PI3K and m-Tor inhibitors. Importantly, proteasome inhibitors were equally effective in exerting toxicity against both mutated and non-mutated cells. Anti-CXCR4 monoclonal antibodies were effective against WM cells, independently of their mutational status [START_REF] Roccaro | G/CXCR4 acts as a driver mutation of tumor progression and modulator of drug resistance in lymphoplasmacytic lymphoma[END_REF]. In the same way, Cao and colleagues have demonstrated ibrutinib resistance among MW cells engineered with CXCR4 mutation [33•]. These data were in support of initial data indicating that CXCR4-mutated patients presented with resistance to ibrutinib but these findings were based on a small cohort of patients (21 mutated patients) [15•, 16]. Secondarily, no response differences were described in international trial iNNOVATE for 38 mutated patients treated with ibrutinib plus rituximab [START_REF] Dimopoulos | Ibrutinib for patients with rituximab-refractory Waldenström's macroglobulinaemia (iNNOVATE): an open-label substudy of an international, multicentre, phase 3 trial[END_REF][START_REF] Dimopoulos | Phase 3 trial of ibrutinib plus rituximab in Waldenström's macroglobulinemia[END_REF]. In the same way, CXCR4 mutational status did not seem to impact statistically rituximab response rate in the rituximab plus placebo group although patients with MyD88 L265P CXCR4 WHIM genotype had a numerically higher rate of major response. More recently, 30 patients with treatment-naive WM were treated with ibrutinib [35•]. A higher incidence of adenopathies was observed among patients with CXCR4 WT disease (n = 16), as described previously. There was a trend for lower IgM level for WT patients but it did not reach statistical significance. Very good partial response rates were higher (31% versus 7%), but not statistically significant, in CXCR4 WT patients. Very interestingly, the median time to major response was delayed in mutated patients and was 7.3 versus 1.8 months (p = 0.01) when compared with WT patients. Similar response kinetics were also previously observed [START_REF] Treon | MYD88 mutations and response to ibrutinib in Waldenström's macroglobulinemia[END_REF][START_REF] Dimopoulos | Ibrutinib for patients with rituximab-refractory Waldenström's macroglobulinaemia (iNNOVATE): an open-label substudy of an international, multicentre, phase 3 trial[END_REF]. Finally, the presence of CXCR4 mutation did not impact on disease response in the recent report of the FILO French group about 69 patients treated with BR [START_REF] Laribi | Bendamustine plus rituximab in newly-diagnosed Waldenstrom macroglobulinemia patients. A study on behalf of the French Innovative Leukemia Organization (FILO)[END_REF]. More studies are needed to conclude about clinical impact of CXCR4 mutation when starting a treatment. At this moment, therapeutic choice could not be based on CXCR4 mutational status, but ibrutinib should be avoided for mutated patients who need urgent efficacy. Interestingly, the development of BTK or PLCG2 mutations was identified only in WM patients with CXCR4 mutations, supporting genomic instability as the basis of clonal evolution.

CXCR4 Mutation and Targeted Therapy

Monoclonal antibody therapy is frequently used in WM. Anti-CD20 antibody rituximab is the most commonly used with great efficacy. New monoclonal antibodies have also been shown to be safe and effective, as anti-CXCR4 antibody. Indeed, targeting the CXCR4-CXCL12 axis with CXCR4 antagonist represents an interesting therapeutic approach in WM patients. Ulocuplumab is a fully humanized anti-CXCR4 monoclonal antibody, with halflife longer than the CXCR4 antagonist plerixafor. It has been shown, in preclinical studies, not only to inhibit CXCR4 activation and migration but also to induce apoptosis of CLL cells at nanomolar concentrations, in the presence or absence of stroma support [START_REF] Kashyap | Ulocuplumab (BMS-936564 / MDX1338): a fully human anti-CXCR4 antibody induces cell death in chronic lymphocytic leukemia mediated through a reactive oxygen speciesdependent pathway[END_REF]. It has been used in association, in clinical trials for the treatment of relapsed/refractory multiple myeloma and acute myeloid leukemia patients, with encouraging results [START_REF] Ghobrial | Phase Ib study of the novel anti-CXCR4 antibody ulocuplumab (BMS-936564) in combination with lenalidomide plus low-dose dexamethasone, or with bortezomib plus dexamethasone in subjects with relapsed or refractory multiple myeloma[END_REF][START_REF] Zhang | Targeting acute myeloid leukemia with a new CXCR4 antagonist IgG1 antibody (PF-06747143)in NOD/SCID mice[END_REF].

In the context of WM, ulocuplumab has been recently studied both in vitro and in vivo and was equally effective in targeting both CXCR4 WHIM and un-mutated WM cells [START_REF] Roccaro | G/CXCR4 acts as a driver mutation of tumor progression and modulator of drug resistance in lymphoplasmacytic lymphoma[END_REF]. The use of the antibody led to significant tumor reduction.

A clinical trial examining ibrutinib with ulocuplumab in symptomatic patients with CXCR4 mutations with WM is ongoing (NCT03225716).

TP53 Alterations: Does It Influence Prognosis and Treatment Response?

TP53 is located at the chromosome 17p13 locus and encodes for a tumor-suppressor protein P53. P53 is a transcription factor and acts as a critical regulator of cellular proliferation, cell-cycle check-point, DNA repair, and apoptosis. In the past 5 years, numerous genomic studies identified TP53 as the most-frequently mutated gene across a large spectrum of cancer types [START_REF] Sabapathy | Therapeutic targeting of p53: all mutants are equal, but some mutants are more equal than others[END_REF]. In lymphoid malignancies, the frequency of TP53 mutations is lower than in other types of cancer but is widely described [START_REF] Xu-Monette | Dysfunction of the TP53 tumor suppressor gene in lymphoid malignancies[END_REF]. The deletion of the short arm of the chromosome 17 (Del17p) is also responsible of defective p53 pathway. The responsiveness of many tumor cells to anticancer therapy-chemotherapy or irradiation-generally seems to be highly dependent on the apoptosis-inducing ability of p53. For this reason, TP53 mutations and del17p are associated with unfavorable therapeutic responses and poor prognosis in many cancers. These mutations have been well described in numerous lymphoid disorders, including CLL and marginal zone lymphoma (MZL) [START_REF] Marinelli | Identification of molecular and functional patterns of p53 alterations in chronic lymphocytic leukemia patients in different phases of the disease[END_REF][START_REF] Parry | Genetics and prognostication in splenic marginal zone lymphoma: revelations from deep sequencing[END_REF]. BTK inhibitors, such as ibrutinib, can be proposed to bypass p53 defective pathway for the treatment. In patients with treatmentnaïve MW, del17p are encountered in 7% of patients and are associated with shorter PFS (18.7 months versus 30 months) and shorter disease-free survival (7.8 months versus 28.8 months) [START_REF] Nguyen-Khac | Chromosomal aberrations and their prognostic value in a series of 174 untreated patients with Waldenström's macroglobulinemia[END_REF]. Recently, TP53 mutations incidence and their functional consequences were analyzed in a series of 125 WM patients [44•]. TP53 mutations were observed in 7.3% of patients and were associated with TP53 deletions in 58.3% of the cases, suggesting a high correlation between TP53 mutations and del17p and a frequent biallelic inactivation of TP53 in WM. TP53 alterations (mutation, deletion, or uniparental disomy) presented a greater frequency of genomic aberrations compared with TP53 WT. Low frequency of TP53 mutation at diagnosis was confirmed by other groups [START_REF] Gustine | TP53 mutations are associated with mutated MYD88 and CXCR4, and confer an adverse outcome in Waldenström macroglobulinaemia[END_REF][START_REF] Varettoni | Pattern of somatic mutations in patients with Waldenström macroglobulinemia or IgM monoclonal gammopathy of undetermined significance[END_REF]. Patients with TP53 alterations had a shorter overall survival and a shorter time to progression to a symptomatic disease. The effect of ibrutinib was assessed in vitro among 8 primary WM cells, of which 3 had TP53 mutation. The sensitivity to induction of cell death was similar independently of TP53 mutational status. Very recently, Gustine et al. have also reported 3 patients with TP53 mutation successfully treated with ibrutinib [START_REF] Gustine | TP53 mutations are associated with mutated MYD88 and CXCR4, and confer an adverse outcome in Waldenström macroglobulinaemia[END_REF]. These data, consistent with observations in other lymphoid disorders, have to be confirmed in vivo with a large number of patients. In case of similar results in vivo, treatment that either bypass the p53 defective pathway (as BTK inhibitors) or reactivate the p53 protein should be considered for patients with TP53 alterations. Ibrutinib may therefore represent an optimal therapeutic option for bypassing the defective TP53 pathway and overcoming chemoresistance in TP53-mutated patients. This has to be confirmed in clinical trials.

At this time, and pending further data, we can conclude that conventional rituximab-based chemotherapies should be preferred in case of MyD88 non-mutated MW or in case of CXCR4 mutation MW with urgent efficacy need.

Conclusion

Although WM is a chronic lymphoproliferative disorder, it remains an incurable disease with risk of mortality. Rituximab-based regimens with chemotherapy have been the most used treatments and new treatment strategies are needed. From a biological point of view, WM is a new oncogenic model with MYD88 L265P mutation as a key driver. Other mutations such as CXCR4 or TP53 may play a role in disease pathogenesis, prognostication in new frontline therapy era and even underlying mechanisms of drug resistance. A first step toward a personalized medicine in WM is currently studied in clinical trials.

Fig. 1

 1 Fig. 1 High-throughput whole-genome sequencing, single nucleotide polymorphism (SNP), and comparative genomic hybridization (CGH) array have identified several pathways characterized with recurrent genomic alterations in WM. Several mechanisms of action are involved in the gene deregulation of key signaling pathways, including deletion, mutation, acquired uniparental disomy (UPD), and gain of chromosome.

Authors' Contribution MB and LS contributed in writing the manuscript; VL and SP contributed in revising the manuscript.

Compliance with Ethical Standards

Conflict of Interest Marine Baron declares that she has no conflict of interest.

Laurence Simon declares that he has no conflict of interest. Stéphanie Poulain declares that she has no conflict of interest. Véronique Leblond has received research funding from Roche, and has received compensation from Roche, Janssen, Gilead, AbbVie, and Servier for service as a consultant.

Human and Animal Rights and Informed Consent This article does not contain any studies with human or animal subjects performed by any of the authors.

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