Friedrich Wehrung
email: friedrich.wehrung01@unicaen.fr

Lmno

GCD-MONOIDS ARISING FROM HOMOTOPY GROUPOIDS

Keywords: 2010 Mathematics Subject Classification. 06F05, 18B35, 18B40, 20L05, 55Q05 Monoid, group, groupoid, category, conical, cancellative, gcd-monoid, simplicial complex, chain complex, barycentric subdivision, universal monoid, universal group, interval monoid, homotopy groupoid, spindle, highlighting expansion

The interval monoid Υ(P ) of a poset P is defined by generators [x, y], where x ≤ y in P , and relations

It embeds into its universal group Υ ± (P ), the interval group of P , which is also the universal group of the homotopy groupoid of the chain complex of P . We prove the following results:

• The monoid Υ(P ) has finite left and right greatest common divisors of pairs (we say that it is a gcd-monoid) iff every principal ideal (resp., filter) of P is a join-semilattice (resp., a meet-semilattice).

• For every group G, there is a connected poset P of height 2 such that Υ(P ) is a gcd-monoid and G is a free factor of Υ ± (P ) by a free group. Moreover, P can be taken to be finite iff G is finitely presented. • For every finite poset P , the monoid Υ(P ) can be embedded into a free monoid. • Some of the results above, and many related ones, can be extended from interval monoids to the universal monoid Umon(S) of any category S. This enables us, in particular, to characterize the embeddability of Umon(S) into a group, by stating that it holds at the hom-set level.

We thus obtain new easily verified sufficient conditions for embeddability of a monoid into a group. We illustrate our results by various examples and counterexamples.

1. Introduction 1.1. Background and highlights. The results of the present paper are, basically, easy. Nonetheless, the purpose they fill is non-trivial, as they lay the foundation for the study of a class of monoids illustrating (cf. Dehornoy and Wehrung [START_REF] Dehornoy | Multifraction reduction III: The case of interval monoids[END_REF]) the limitations of an approach, initiated in Dehornoy [START_REF]Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC[END_REF], of the word problem in certain groups. The class of groups in question includes Artin-Tits groups, and, more generally, universal groups of cancellative monoids with left and right greatest common divisors -in short gcd-monoids (cf. Dehornoy [START_REF]Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC[END_REF]).

Those monoids, together with the following construct, are our main object of study. The interval monoid Υ(P ) of a poset (i.e., partially ordered set) P is the universal monoid of the category associated with P in the usual manner, and its universal group Υ ± (P ) is also the universal group of the homotopy groupoid of the chain complex of P . Accordingly, we will call Υ ± (P ) the floating homotopy group of P (cf. Definition 6.1).

The monoids Υ(P ) are fairly special objects: for example, Υ(P ) always embeds into its universal group Υ ± (P ) (cf. Proposition 7.6), and further, if P is finite, then Υ(P ) always embeds into a free monoid (Proposition 8.2). Nevertheless, every group is a free factor, by a free group, of the universal group of a gcd-monoid of the form Υ(P ) (see Theorem 7.10 below).

The concept of a gcd-monoid, which is implicit in the approach of braids and Artin-Tits monoids by Garside [START_REF] Garside | The braid group and other groups[END_REF] and Brieskorn-Saito [START_REF] Brieskorn | Artin-Gruppen und Coxeter-Gruppen[END_REF], and underlies all that is now called Garside theory (cf. Dehornoy [START_REF]Foundations of Garside Theory[END_REF]), was introduced as such in Dehornoy [START_REF]Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC[END_REF].

The present paper sets the foundation of the study of the interaction between groups, gcd-monoids, and interval monoids. Three highlights are the following.

Theorem 5.9. The universal monoid of a category S is a gcd-monoid iff S is cancellative, has no non-trivial left invertibles, and has left gcds (resp., right gcds) of any pair of elements with the same source (resp., target ).

In particular, for a poset P , the condition that Υ(P ) be a gcd-monoid can be easily read on P , by saying that every principal ideal is a join-semilattice and every principal filter is a meet-semilattice (Proposition 7.8). This condition holds if P is the barycentric subdivision of a simplicial complex (Corollary 7.9). Theorem 7.10. Every group G is a free factor, by a free group, of the universal group Υ ± (P ) of Υ(P ), for some poset P of height 2 such that Υ(P ) is a gcd-monoid. Furthermore, P can be taken to be finite iff G is finitely presented.

Theorem 10.1. The universal monoid of a category S embeds into a group iff there is a functor, from S to a group, which separates every hom-set of S.

1.2. Section by section summary of the paper. We start by recalling, in Section 2, the basic concepts and notation underlying the paper.

For any category S, described in an "arrow-only" fashion via its partial semigroup of arrows, the universal monoid U mon (S) of S is the initial object in the category of all functors from S to some monoid. The universal group U gp (S) of S is defined similarly for groups. In Section 3, we present an overview of those concepts, essentially originating in Higgins [START_REF] Higgins | Notes on Categories and Groupoids[END_REF].

Say that a category is conical if it has no non-trivial left (equivalently, right) invertibles. The main goal of Section 4 is to prove that a category S is conical (resp., cancellative) iff the monoid U mon (S) is conical (resp., cancellative).

In Section 5, we describe the left and right divisibility preorderings on U mon (S) and we prove that U mon (S) is a gcd-monoid iff S has left (resp., right) gcds of pairs with the same source (resp., target).

In Section 6, we study the "floating homotopy group Υ ± (K)" of a simplicial complex K, observing in particular that it is the universal group of the homotopy groupoid Π 1 (K) of K. We observe in Proposition 6.3 that for a spanning tree with set of edges E in a connected simplicial complex K, Υ ± (K) is the free product of the fundamental group of K and the free group on E.

The interval monoid Υ(P ), of a poset P , is defined, in Section 7, as the universal monoid of the category Cat(P ) canonically associated to P (i.e., there is exactly one arrow from x to y iff x ≤ y, no arrow otherwise). We verify that the universal group Υ ± (P ) of Υ(P ) is identical to the floating homotopy group Υ ± (Sim(P )) of the chain complex Sim(P ) of P , thus showing that there is no ambiguity on the notation Υ ± (P ). We also verify that the canonical map from Υ(P ) to Υ ± (P ) is one-to-one -of course this result does not extend to arbitrary categories. We characterize those posets P such that Υ(P ) is a gcd-monoid, and we show that this condition always holds if P is the barycentric subdivision of a simplicial complex.

In Section 8 we verify that Υ(P ) always embeds into a free monoid (not only a free group) if P is finite, but that nonetheless, we construct a submonoid of the free monoid on four generators, which is a gcd-monoid, but not the interval monoid of any poset.

In Section 9, we introduce a class of monoids, denoted by Υ(P, u, v), where [u, v] is a so-called extreme spindle of a poset P . The monoids Υ(P, u, v) are also constructed as universal monoids of categories, and they are used as counterexamples in the follow-up paper Dehornoy and Wehrung [START_REF] Dehornoy | Multifraction reduction III: The case of interval monoids[END_REF]. A presentation of Υ(P, u, v) consists of a certain subset of the natural presentation of Υ(P ) (cf. Proposition 9.6), and if Υ(P ) is a gcd-monoid, then so is Υ(P, u, v) (cf. Proposition 9.5).

In Section 10 we show that for any category S, the universal monoid U mon (S) can be embedded into a group iff the embedding can be realized at the hom-set level. We illustrate that result on an example, which we denote by C 6 (Example 10.2). While neither Adjan's condition nor Dehornoy's 3-Ore condition are a priori applicable to prove the embeddability of C 6 into a group, this example is the universal monoid of a category, so the above-mentioned criterion easily applies.

Notation and terminology

The present paper will involve small categories, viewed in an "arrow-only" fashion as partial semigroups with identities subjected to certain axioms (cf. Definition 3.1), as well as categories viewed in the usual object / arrow way, which are then usually proper classes. In order to highlight the distinction without dragging along the qualifier "small" through the paper, we will denote the first kind (small categories, usually arrow-only) as "categories", and the second kind (object / arrow categories) as "Categories". Similar conventions will apply to groupoids versus Groupoids (categories where every arrow is an isomorphism) and functors versus Functors (morphisms of categories).

The following Categories will be of special importance:

• Cat, the Category of all categories (the morphisms are the functors),

• Gpd, the Category of all groupoids (the morphisms are the functors),

• Mon, the Category of all monoids (the morphisms are the monoid homomorphisms), • Gp, the Category of all groups (the morphisms are the group homomorphisms), • Pos, the Category of all posets (i.e., partially ordered sets; the morphisms are the isotone maps, that is, those maps f : P → Q such that x ≤ y implies f (x) ≤ f (y) whenever x, y ∈ P ). Seq X denotes the set of all (possibly empty) finite sequences of elements of any set X. Denote by x y the concatenation of finite sequences x and y. For a finite sequence x = (x 1 , . . . , x n ), we set n = lh(x), the length of x (thus equal to 0 for the empty sequence). We will often use the notational convention x = (x 1 , . . . , x n ) , for x ∈ Seq X , where n = lh(x) .

(2.1)

As in Rotman [START_REF]An Introduction to the Theory of Groups[END_REF]Chapter 11], a simplicial complex (or abstract simplicial complex ) K is a collection of nonempty finite subsets, called simplices, of a set Vert K, the vertices of K, such that every {v}, where v ∈ Vert K, is a simplex, and every nonempty subset of a simplex is a simplex. The n-skeleton of K, denoted by K (n) , is the set of all simplices with at most n + 1 elements (also called n-simplices), for each n ∈ N. We say that K has dimension at most

n if K = K (n) . A simplicial complex K ′ is a subcomplex of K if every simplex of K ′ is a simplex of K. A subcomplex K ′ of K is spanning if it has the same vertices as K.
A path, in K, is defined as a nonempty finite sequence x = (x 0 , . . . , x n ), where each x i ∈ Vert K and each {x i , x i+1 } ∈ K (1) . Set ∂ 0 x = def x 0 and ∂ 1 x = def x n , and say that x is a path from x 0 to x n . The homotopy relation, on the set Path(K) of all paths of K, is the equivalence relation ≃ on Path(K) generated by all pairs

u (x, x) v ≃ u (x) v , where u (x) v is a path, u (x, y, z) v ≃ u (x, z) v ,
where u (x, y, z) v is a path and {x, y, z} is a simplex .

Paths x = u (z) and y = (z) v can be multiplied, by setting xy = u (z) v (not to be confused with the concatenation x y), and this partial operation defines a category structure on Path(K), whose identities are the one-entry paths of K. We say that K is connected if there is a path between any two vertices of K. Observe that x ≃ y implies that ∂ 0 x = ∂ 0 y and ∂ 1 x = ∂ 1 y, for all paths x and y. Denoting by

[x] = [x 0 , . . . , x n ], or [x] K = [x 0 , . . . ,
x n ] K in case K needs to be specified, the homotopy class of a path x = (x 0 , . . . , x n ), homotopy classes can be multiplied,

via the rule [x] • [y] = [xy], which is defined iff ∂ 1 x = ∂ 0 y. The collection Π 1 (K)
of all homotopy classes of paths of K is a groupoid, called the edge-path groupoid, or fundamental groupoid, of K. The inverse [x] -1 of a homotopy class [x], where x = (x 0 , . . . , x n ), is the homotopy class of x -1 = def (x n , . . . , x 0 ).

For any poset P , the chain complex Sim(P ) of P has vertices the elements of P , and simplices the finite chains of P . An element x of P is a lower cover of an element y of P , in notation x ≺ y, if x < y and there is no z such that x < z < y.

For any simplicial complex K, the barycentric subdivision of K is a poset, defined as the set of all simplices of K, partially ordered under set inclusion.

The height of a poset P is defined as the supremum of the cardinalities of all chains of P , minus one.

We say that a group G is a free factor (resp., doubly free factor ) of a group H, if there exists a group (resp., a free group) F such that H ∼ = F * G, where * denotes the free product (i.e., coproduct) in the Category Gp of all groups.

We denote by F mon (X) (resp., F gp (X)) the free monoid (resp., group) on a set X. We denote by N = {0, 1, 2, . . . } the additive monoid of all nonnegative integers, and by Z the additive group of all integers.

The universal monoid of a category

The universal monoid construction, applied to a category S, is a special case of a construction described in Chapters 8 to 10 of Higgins [START_REF] Higgins | Notes on Categories and Groupoids[END_REF]. Its underlying monoid is obtained by keeping all existing products in S and collapsing all the identities of S. Although most of the material in this section is contained in some form in Higgins [START_REF] Higgins | Notes on Categories and Groupoids[END_REF], we will write it in some detail, in order to be able to apply it to the category-to-monoid transfer results of Sections 4 and 5.

For a partial binary operation • on a set S, we will abbreviate the statement that x • y is defined (resp., undefined) by writing x • y ↓ (resp., x • y ↑), for any x, y ∈ S. Hence, x•y ↓ is equivalent to the statement (∃z)(z = x•y). This definition is extended the usual way to arbitrary terms, of the language of semigroups, with parameters in a given semigroup S: for example, for any x, y, z ∈ S, (x • y) • z ↓ holds if there are u, v ∈ S such that u = x • y and v = u • z; and t ↑ means that the term t is undefined (in S). Definition 3.1. A semicategory is a structure (S, •), consisting of a set S endowed with a partial binary operation

• such that x • (y • z) ↓ iff (x • y) • z ↓ iff x • y ↓ and y • z ↓, and then x • (y • z) = (x • y) • z (sometimes denoted by x • y • z,
and usually by xyz), for all x, y, z ∈ S.

An element e ∈ S is an identity of S if it is idempotent (i.e., e 2 = e) and xe ↓ implies that xe = x and ex ↓ implies that ex = x, for all x ∈ S. We denote by Idt S the set of all identities of S.

A category is a semicategory in which for every x (thought of as an arrow) there are (necessarily unique) identities a and b such that x = ax = xb. We will write a = ∂ 0 x (the source of x) and b = ∂ 1 x (the target of x). The hom-sets of S are then the S(a, b)

= {x ∈ S | ∂ 0 x = a and ∂ 1 x = b}, for a, b ∈ Idt S.
For categories S and T , a map f : S → T is a functor if f (xy) = f (x)f (y) whenever xy is defined, and f sends identities to identities. In particular, if T is a monoid (i.e., a category with exactly one identity, then denoted by 1), f should send every identity to 1.

Of course, if a category S is described, in the usual fashion, by its objects and morphisms, the identities of S are exactly the identity morphisms on its objects. We will sometimes write ∂ 0 a a → ∂ 1 a, whenever a ∈ S. Definition 3.2. Let S be a category. An element a ∈ S is • right invertible if there exists x ∈ S (which is then called a right inverse of a) such that ax is an identity -then of course, necessarily, ax = ∂ 0 a; • left invertible if there exists x ∈ S (which is then called a left inverse of a)

such that xa is an identity -then of course, necessarily, xa = ∂ 1 a; • invertible if it is both left and right invertible; • left cancellable (or monic), if ax = ay implies that x = y, for all x, y ∈ S; • right cancellable (or epic), if xa = ya implies that x = y, for all x, y ∈ S; • cancellable if it is both right and left cancellable.

The category S is left cancellative (right cancellative, cancellative, respectively) if every element of S is left cancellable (right cancellable, cancellable, respectively). A groupoid is a category in which every element is invertible.

From now on until Proposition 3.12 we shall fix a category S. For a, b ∈ Seq S, we say that a reduces to b in one step, in notation a → b, if there are u, v ∈ Seq S such that either there is e ∈ Idt S such that a = u (e) v and b = u v, or there are x, y ∈ S such that xy ↓ , a = u (x, y) v, and b = u (xy) v.

Obviously, a → b implies that lh(a) = lh(b) + 1. We denote by → * the reflexive and transitive closure of →, and we say that a reduces to b if a → * b. For all x, y ∈ Seq S, let x ≡ y hold if there is z ∈ Seq S such that x → * z and y → * z. (3) Both relations → and → * are compatible with concatenation, that is, x → y (resp., x → * y) implies that x z → y z and z x → z y (resp., x z → * y z and z x → * z y), for all x, y, z ∈ Seq S. (4) The binary relation ≡ is a monoid congruence on (Seq S, ).

Proof. Ad [START_REF] Sergeȋ | Defining relations and algorithmic problems for groups and semigroups[END_REF]. The problem reduces to a small number of cases, all of which are easy, and the only three of which not being completely trivial (e.g., requiring both definitions of a semicategory and an identity) being the following:

a = u (x, e) v , b 0 = u (x) v , b 1 = u (xe) v ,
where u, v ∈ Seq S , x ∈ S , e ∈ Idt S , and xe ↓ ,

a = u (e, x) v , b 0 = u (x) v , b 1 = u (ex) v ,
where u, v ∈ Seq S , x ∈ S , e ∈ Idt S , and ex ↓ , and

a = u (x, y, z) v , b 0 = u (x, yz) v , b 1 = u (xy, z) v ,
where u, v ∈ Seq S , x, y, z ∈ S , xy ↓ , and yz ↓ .

In the first two cases, b 0 = b 1 . In the third case, take c = u (xyz) v.

It is well known that (2) follows from [START_REF] Sergeȋ | Defining relations and algorithmic problems for groups and semigroups[END_REF].

(3) is straightforward, and ( 4) is an easy consequence of (2) and [START_REF] Brieskorn | Artin-Gruppen und Coxeter-Gruppen[END_REF].

From now on we shall set, following the convention set in (2.1),

Seq red S = def {x ∈ Seq(S \ Idt S) | x i x i+1 ↑ whenever 1 ≤ i < i + 1 ≤ lh(x)} ,
the set of all (finite) reduced sequences of elements of S. Obviously, a finite sequence x ∈ Seq S is reduced iff there is no y ∈ Seq S such that x → y, iff x is maximal with respect to the partial ordering → * . Observe that larger words, with respect to → * , have smaller length.

Lemma 3.4. For each x ∈ Seq S, there exists a unique element of Seq red S, which we shall denote by x red , such that x → * x red . Furthermore, x red can be characterized in each of the following three ways:

• x red is the largest element of x/≡ with respect to the partial ordering → * ; • x red is the unique element of x/≡ with smallest length; • x red is the unique reduced sequence equivalent to x modulo ≡.

Proof. The existence statement follows trivially from the fact that → * decreases the length. The uniqueness statement follows trivially from Lemma 3.3. Then the proof of the equivalence of the three statements about x red is straightforward.

The last observation of Lemma 3.4 is, essentially, contained (with a different proof) in Theorem 4 of Chapter 10 in Higgins [START_REF] Higgins | Notes on Categories and Groupoids[END_REF] (take A = S and define σ as the constant map with value 1). Notation 3.5. We denote by U mon (S) the quotient monoid (Seq S)/≡. Moreover, we denote by x/≡ the ≡-equivalence class of a finite sequence x ∈ Seq S, and we set ℓ(x) = def lh x red .

The elements of U mon (S) are exactly the equivalence classes x/≡, where x ∈ Seq S. In particular, the unit element of U mon (S), which we shall denote by 1, is the ≡-equivalence class of the empty sequence ∅.

Since every equivalence class x/≡ is uniquely determined by x red , which is its representative of smallest length (cf. Lemma 3.4), the monoid U mon (S) can be alternatively described as consisting of all the reduced finite sequences (i.e., Seq red S), endowed with the multiplication defined by x • S y = (x y) red , for all x, y ∈ Seq red S .

( The following result shows that the description of the multiplication given in (3.1) is especially convenient in case S is conical. Lemma 3.7. Let the category S be conical. Then the multiplication given by (3.1) is characterized by

1 • S x = x • S 1 = x, for any x ∈ U mon (S), together with u (x) • S (y) v = u (x, y) v , if xy ↑ u (xy) v , if xy ↓ , for all u (x), (y) v ∈ Seq red S . (3.2)
Proof. Suppose that xy is defined. Since S is conical and neither x nor y is an identity, xy is not an identity, thus the finite sequence u (xy) v remains reduced in the second case of (3.2).

In order to ease notation, we will usually write xy, or sometimes x • y in case we need a separator, instead of x • S y.

A straightforward application of Lemma 3.7 yields the following corollary.

Corollary 3.8. The following statements hold, for any category S:

(1) ℓ(xy) ≤ ℓ(x) + ℓ(y), for all x, y ∈ U mon (S). (2) If, in addition, S is conical, then (i) ℓ(x) ≤ ℓ(xy) and ℓ(y) ≤ ℓ(xy); (ii) ℓ(xy) ∈ {ℓ(x) + ℓ(y) -1, ℓ(x) + ℓ(y)}, for all x, y ∈ Seq red S.
We will usually work with the description of U mon (S) as Seq red S endowed with the multiplication defined by (3.1). A noticeable exception, involving the description of U mon (S) as (Seq S)/≡ , is the following straightforward universal characterization of U mon (S).

Proposition 3.9. The map ε S : S → U mon (S), x → (x)/≡ is a functor. Furthermore, the pair (U mon (S), ε S ) is an initial object in the Category of all functors from S to a monoid. Hence, U mon defines a left adjoint of the inclusion Functor Mon ֒→ Cat.

Accordingly, we shall call the monoid U mon (S) (or, more precisely, the pair (U mon (S), ε S )) the universal monoid of S.

The requirement that ε S be a functor -thus collapses all identities -implies that ε S is not one-to-one as a rule (so S may not be a partial subsemigroup of U mon (S)). Nevertheless, it falls short of being so. The following observation is a specialization, to the construction U mon (S), of Corollary 1 of Chapter 10 in Higgins [START_REF] Higgins | Notes on Categories and Groupoids[END_REF].

Lemma 3.10. For all x, y ∈ S, ε S (x) = ε S (y) iff either x = y or x and y are both identities. In particular, the restriction of ε S to every hom-set of S is one-to-one.

Proof. It is obvious that (x) red is equal to (x) if x is not an identity, and the empty sequence otherwise. The first statement follows immediately. Now let a, b ∈ Idt S and let x, y ∈ S(a, b) such that ε S (x) = ε S (y). If x = y, then, by the paragraph above, x = y = a, a contradiction. Notation 3.11. We denote by U gp (M ) the universal group of a monoid M -that is, the initial object in the Category of all monoid homomorphisms from M to a group. We extend this notation to all categories, by setting

U gp (S) = def U gp (U mon (S))
for every category S.

Pre-composing the canonical monoid homomorphism U mon (S) → U gp (S) with the canonical homomorphism ε S : S → U mon (S) yields the canonical homomorphism η S : S → U gp (S), and the following easy consequence of Proposition 3.9.

Proposition 3.12. The map η S : S → U gp (S) is a group-valued functor. Furthermore, the pair (U gp (S), η S ) is an initial object in the Category of all functors from S to a group. Hence, U gp defines a left adjoint of the inclusion Functor Gp ֒→ Cat.

Accordingly, we shall call the group U gp (S) (or, more precisely, the pair (U gp (S), η S )) the universal group of S. Remark 3.13. For any category S, every element x ∈ U mon (S) can be written in a unique way as a product ε S (x 1 ) • • • ε S (x n ), which we will often abbreviate as x 1 • • • x n , where no x i is an identity and no product x i x i+1 is defined: namely, let x red = (x 1 , . . . , x n ). The elements of S \ Idt S, here identified with their images under ε S , will be called the standard generators of the monoid U mon (S).

In particular, if S is a groupoid, then every element of U mon (S) is a product of invertibles, thus is itself invertible; and thus U mon (S) = U gp (S) is the universal group of S.

Preservation of conicality and cancellativity

The main goal of this section is to prove that conicality and cancellativity of a category can be read on its universal monoid. The case of conicality is easy. Observing that ℓ(x) = 0 iff x = 1, for any x ∈ U mon (S), it follows immediately from Corollary 3.8 that if S is conical, then so is U mon (S).

Preservation of cancellativity and invertibility are slightly less straightforward.

Lemma 4.2. The following statements hold, for any element a in a category S:

(1) a is left cancellable in S iff ε S (a) is left cancellable in U mon (S); (2) a is right cancellable in S iff ε S (a) is right cancellable in U mon (S); (3) a is right invertible in S iff ε S (a) is right invertible in U mon (S); (4) a is left invertible in S iff ε S (a) is left invertible in U mon (S).
Proof. We prove the results about left cancellativity and right invertibility. The results about right cancellativity and left invertibility follow by symmetry.

Let ε S (a) be left cancellable in U mon (S) and let x, y ∈ S such that ax = ay. In particular, x and y have the same source and the same target. Moreover, since ε S (a)ε S (x) = ε S (a)ε S (y) and ε S (a) is left cancellable in U mon (S), we get ε S (x) = ε S (y). By the second part of Lemma 3.10, it follows that x = y.

Suppose, conversely, that a is left cancellable in S. We need to prove that ε S (a) is left cancellable in U mon (S).

Suppose first that a is right invertible in S and let s be a right inverse of a in S. From as = ∂ 0 a it follows that asa = a = a∂ 1 a, thus, since a is left cancellable in S, sa = ∂ 1 a. It follows that a is invertible in S, so ε S (a) is invertible in U mon (S) and the desired conclusion holds.

Suppose from now on that a is not right invertible in S. For every reduced sequence z = (z 1 , . . . , z k ) ∈ U mon (S), two cases can occur:

• either k = 0 or az 1 is undefined. Then ε S (a)z = (a, z 1 , . . . , z k ).

• k > 0 and az 1 is defined. Then ε S (a)z = (az 1 , z 2 , . . . , z k ).

The putative "third case" suggested by the above, namely k > 0 and az 1 is an identity of S, does not occur, because a is not right invertible in S. In particular, observe the following: Let us now suppose that a is right invertible in S, that is, as = ∂ 0 a for some s ∈ S. Then ε S (a)ε S (s) = 1, thus ε S (a) is right invertible in U mon (S).

ℓ ε S (a)z ∈ {ℓ(z), ℓ(z) + 1} . ( 4
Suppose, finally, that ε S (a) is right invertible in U mon (S), we must prove that a is right invertible in S. Suppose otherwise. By assumption, there is a reduced se-

quence b = (b 1 , . . . , b n ) ∈ U mon (S) such that ε S (a)b = 1. In particular, (a, b 1 , . . . , b n ) is reducible,
thus, since a is not an identity and b is a reduced sequence, n > 0 and ab 1 is defined. Since (ab 1 , b 2 , . . . , b n ) should be equivalent to the empty sequence, it is reducible, thus ab 1 is an identity, a contradiction.

Since every element of U mon (S) is a finite product of elements of the range of ε S , and since left cancellativity, right cancellativity, and invertibility are all preserved under finite products, we obtain immediately the following.

Corollary 4.3. A category S is left cancellative (resp., right cancellative) iff its enveloping monoid U mon (S) is left cancellative (resp., right cancellative). Moreover, S is a groupoid iff U mon (S) is a group.

Gcd-categories and gcd-monoids

In this section we extend, to an arbitrary category, Dehornoy's definition of a gcd-monoid, and we prove, in the same spirit as in Section 4, that a category is a gcd-category iff its universal monoid is a gcd-monoid. Definition 5.1. Every category S carries two partial preorderings S and S , defined by letting a S b (resp., a S b) hold if there exists x ∈ S such that b = ax (resp., b = xa) -we say that a left divides (resp., right divides) b, or, equivalently, that b is a right multiple (resp., left multiple) of a. If such an x is unique, then we write x = b S a (resp., x = b S a). We will call S (resp., S ) the left divisibility preordering (resp., the right divisibility preordering) of S.

Following tradition, the greatest lower bound of a subset X of S if it exists, with respect to S (resp., S ), will be called the left gcd (resp., right gcd ) of X and denoted by S X (resp., S X). Likewise, the least upper bound of a subset X of S if it exists, with respect to S (resp., S ), will be called the right lcm (resp., left lcm) of X and denoted by S X (resp., S X). If X = {a, b}, we will write a ∧ S b, a ∧ S b, a ∨ S b, a ∨ S b, respectively.

As usual, we drop the index S in the notations above in case S is understood.

Observe, in particular, that an element a ∈ S is right invertible (resp., left invertible) iff a S ∂ 0 a (resp., a S ∂ 1 a). Definition 5.2. For a reduced sequence x = (x 1 , . . . , x n ) ∈ U mon (S), with n > 0, the left component ∇ 0 x and the right component ∇ 1 x of x are the elements of S respectively defined as ∇ 0 x = x 1 and ∇ 1 x = x n .

Although it is tempting to extend Definition 5.2 by setting ∇ 0 1 = ∇ 1 1 = 1, a new "unit" element, this element 1 would need to live outside the category S, and the new structure S ∪ {1} would usually no longer be a category, which could open the way to a whole collection of incidents. We therefore chose to restrict the domain of definition of both ∇ 0 and ∇ 1 to U mon (S) \ {1}.

The following result states that the component maps ∇ 0 and ∇ 1 are right adjoints of the canonical map ε S : S → U mon (S) with respect to the divisibility preorderings on S and U mon (S).

Lemma 5.3. Let S be a category and let a ∈ S.

Set M = U mon (S). Then ε S (a) M 1 iff a S ∂ 0 a and ε S (a) M 1 iff a S ∂ 1 a. Furthermore, if a is not right invertible, then ε S (a) M b iff a S ∇ 0 b , (5.1) if a is not left invertible, then ε S (a) M b iff a S ∇ 1 b , (5.2)
for any b ∈ M \ {1}.

c = (c 1 , . . . , c n ) ∈ M such that b = ε S (a)c. Since a is not right invertible, ac 1 is not an identity (if n > 0 and ac 1 ↓), thus we get b = ε S (a)c = (a, c 1 , . . . , c n ) , if n = 0 or ac 1 ↑ , (ac 1 , c 2 , . . . , c n ) , if n > 0 and ac 1 ↓ .
In the first case, ∇ 0 b = a. In the second case, ∇ 0 b = ac 1 . In both cases, we get a S ∇ 0 b. Following the proofs above in reverse yields easily the desired equivalence.

The following technical lemma states in which case a left or right gcd exists in U mon (S), along with an algorithm enabling us to compute that gcd.

Lemma 5.6. Let S be a conical category, set M = U mon (S), let (a i | i ∈ I) be a nonempty family of elements in U mon (S), and let c ∈ M . We denote by a left (resp., a right ) the longest common prefix (resp., suffix ) of the a i . Then the following statements hold: (i) c = a right and either a right = a i for some i or

{∂ 1 ∇ 1 (a i M a right ) | i ∈ I} is not a singleton, or the right gcd of {∇ 1 (a i M a right ) | i ∈ I} is an iden- tity; (ii) a i = a right for all i, the greatest lower bound c of {∇ 1 (a i M a right ) | i ∈ I}
exists in S and it is not an identity, and c = (c) a right .

Proof. By symmetry, it suffices to prove (1) and (2). Since a left M a i for all i ∈ I, c M a left implies that c M a i for all i ∈ I. Now suppose that a = a i for all i ∈ I, and let c ∈ S such that c = a left (c) and c S ∇ 0 (a i M a left ) for each i ∈ I.

Then c = a left • ε S (c) M a left • (a i M a left ) = a i
, for all i ∈ I . Suppose, conversely, that c M a i for all i ∈ I. We prove that the condition given on the right hand side of (1) holds for c. We may assume that c = 1, so c = c * (c) for some c * ∈ M and some c ∈ S \ Idt S. Since S is conical and by Lemma 5.5, ℓ(c) ≤ ℓ(a i ) for all i ∈ I and c * is a proper prefix of all a i , thus also of a left . Write a i = c * (a i,1 , . . . , a i,ni ), for a reduced sequence (a i,1 , . . . , a i,ni ) with n i > 0. Since c = c * (c) M a i , it follows from Lemma 5.5 that c S a i,1 for all i ∈ I .

(5.3) Suppose first that c * is a proper prefix of a left . Then the a i,1 are all equal to the entry of a left following c * , so they form a constant family, thus, by (5.3) together with Lemma 5.5, we get c M a left .

The only remaining possibility is c * = a left , so (since each n i > 0) a left is a proper prefix of each a i . Since each a i = a left (a i,1 , . . . , a i,ni ), we get a i,1 = ∇ 0 (a i M a left ), and thus, by (5.3), c S ∇ 0 (a i M a left ). This completes the proof of (1). Now we prove [START_REF] Bergman | On monoids, 2-firs, and semifirs[END_REF]. The additional assumption that S be left cancellative, together with Proposition 4.1 and Corollary 4.3, ensures that M is also conical and left cancellative. By Lemma 5.4, it follows that M is a partial ordering.

It follows from (1) that the left gcd of {a i | i ∈ I}, if it exists, is either equal to a left or has the form a left (c) for some c. The case where a left = a i , for some i, being trivial, we may assume that a left is a proper prefix of each a i . In that case, by (1), the gcd of If S is a monoid, then we say that S is a left gcd-monoid, right gcd-monoid, gcd-monoid, respectively.

{a i | i ∈ I} is equal to a left iff either the set {∇ 0 (a i M a left ) | i ∈ I} has no left divisor, that is, {∂ 0 ∇ 0 (a i M a left ) | i ∈ I} is not a singleton,
The terminology introduced in Definition 5.8 is consistent with the one of Dehornoy [START_REF]Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC[END_REF] (see Section 1.1 for more background on gcd-monoids). Gcd-monoids and gcd-categories are related by the following result. Theorem 5.9. A category S is a left gcd-category (right gcd-category, gcd-category, respectively) iff its universal monoid U mon (S) is a left gcd-monoid (right gcdmonoid, gcd-monoid, respectively). Let, conversely, S be a left gcd-category and let (a i | i ∈ I) be a nonempty finite family of elements of M . Denote by a left the largest common prefix of the a i . We apply Lemma 5.6 [START_REF] Bergman | On monoids, 2-firs, and semifirs[END_REF]. If either a i = a left for some i, or {∂ 0 ∇ 0 (a i M a left ) | i ∈ I} is not a singleton, or the only left divisor of {∇ 0 (a i M a left ) | i ∈ I} is an identity, then the left gcd of the a i is a left . Now suppose that each a i properly extends a left and that {∇ 0 (a i M a left ) | i ∈ I} has a non-identity left divisor. By assumption and since I is finite nonempty, this means that {∇ 0 (a i M a left ) | i ∈ I} has a nonidentity left gcd, say c, in S. Observe that the sequence a left (c) is reduced. By Lemma 5.6(2), it follows that a left (c) is the left gcd of

{a i | i ∈ I} in M .
The following result extends Corollary 5.7 to lcms. Since we did not define left or right lcm-categories, our assumption (existence of both left and right gcds) needs to be a bit stronger (cf. Dehornoy [6, Lemma 2.14]). Remark 5.11. Lemma 5.3 can be interpreted in terms of normal forms and Garside families, for the monoid U mon (S) (those concepts are defined in Dehornoy [START_REF]Foundations of Garside Theory[END_REF]). Suppose that S is a gcd-category, and let a ∈ U mon (S), written as a reduced sequence, say a = a 1 = (a 1 , . . . , a n ). By Lemma 5.3, the element a 1 = ∇ 0 a is the largest element of S, with respect to left divisibility, left dividing a. Furthermore, since U mon (S) is left cancellative (cf. Corollary 4.3), the reduced sequence a 2 = def (a 2 , . . . , a n ) is the only reduced sequence satisfying a = ε S (a 1 ) • a 2 , and then the above process can be started again on a 2 , yielding a 2 as the largest element of S, with respect to left divisibility, left dividing a 2 ; and so on. At the end of that process, we obtain that the expression a = a 1 • • • a n is the (left) greedy normal form of a, associated with the Garside family ε S [S] (in one-to-one correspondence with (S \ Idt S) ∪ {1}) of U mon (S).

An unusual feature of that Garside family is that it is two-sided : it is closed under both left and right divisors (any divisor of an element is an element), and also, it is closed under finite lcms (cf. Proposition 5.10). The left greedy normal form and the right greedy normal form, with respect to that family, are identical.

Floating homotopy group of a simplicial complex

Although groupoids are instrumental in getting to our results, our main goals are the investigation of certain monoids and groups. This partly explains why we are less interested in the homotopy groupoid Π 1 (K) (where K is a simplicial complex) than its universal group, which we shall denote by Υ ± (K). In the present section we observe, in particular, that if K is connected, then Υ ± (K) is the free product of the fundamental group π 1 (K, -) by the free group on the edges of a spanning tree of K. This yields a convenient recipe for quick calculation of floating homotopy groups. Easy instances of that recipe are given in Examples 6.4 and 7.5. Definition 6.1. The floating homotopy group of a simplicial complex K, denoted by Υ ± (K), is the group defined by generators [x, y] (or [x, y] K in case K needs to be specified), where {x, y} ∈ K (1) , and relations [x, z] = [x, y] • [y, z] , for all x, y, z ∈ Vert K such that {x, y, z} ∈ K (2) . (6.1) An alternative definition of Υ ± (K), suggested to the author by a referee, runs as follows. For a vertex o not in K, the cone o * K of K is defined as the simplicial complex consisting of all simplices of K together with all sets of the form {o} ∪ X where X is either empty or a simplex of K. The edges {o, x}, where x is a vertex of K, define a spanning tree of o * K. By applying Tietze's Theorem (cf. Rotman [START_REF]An Introduction to the Theory of Groups[END_REF]Theorem 11.31]), we obtain the relation Υ

± (K) ∼ = π 1 (o * K, o).
Heuristically, Υ ± (K) is the group universally generated by all homotopy classes of paths in K, with endpoints not fixed -thus our terminology "floating". In fact, it is straightforward to verify the following description of Υ ± (K) in terms of the homotopy groupoid Functor Π 1 (cf. Section 2) and the universal group Functor U gp (cf. Section 3). Proposition 6.2. Υ ± (K) = U gp (Π 1 (K)), for every simplicial complex K.

In particular, for every vertex p ∈ Vert K, the fundamental group π 1 (K, p), of all homotopy classes of all loops at p, is a subgroup (vertex group) of the groupoid Π 1 (K). By Lemma 3.10, π 1 (K, p) is thus a subgroup of Υ ± (K): the homotopy class [x] of a path x = (x 0 , . . . , x n ), with

x 0 = x n = p, is identified with the element [x 0 , x 1 ] • • • [x n-1 , x n ] of Υ ± (K).
Proposition 6.3. Let K be a connected simplicial complex, let E be the set of all edges of a spanning tree of K, and let p be a vertex of K. Then Υ ± (K) ∼ = F gp (E) * π 1 (K, p). Hence, π 1 (K, p) is a doubly free factor of Υ ± (K).

Proof. Denote by F the free groupoid on the graph E (cf. Higgins [START_REF] Higgins | Notes on Categories and Groupoids[END_REF]Chapter 4]). By applying Corollary 1, Chapter 12 in Higgins [START_REF] Higgins | Notes on Categories and Groupoids[END_REF], to the fundamental groupoid Π 1 (K) of K, we obtain the relation Π 1 (K) ∼ = F * π 1 (K, p) (where * denotes the coproduct within the Category of all groupoids). Since the universal group Functor U gp : Gpd → Gp is a left adjoint, it preserves coproducts. By applying it to the relation above, and since U gp (F ) ∼ = F gp (E), we get our result. Example 6.4. Consider the one-dimensional simplicial complex K represented in Figure 6.1. A direct calculation shows that Υ ± (K) is a free group on the edges A spanning tree of K is represented with thick it has three edges. By using Tietze's Theorem, we get π 1 (K, -) ∼ = Z (this is trivial in more than one way: for example, the geometric realization of K is a circle). We get again Υ

± (K) ∼ = Z * F gp (3) ∼ = F gp (4).
See also Example 7.5.

Interval monoid and interval group of a poset

Every poset P gives rise, in a standard way, to a category Cat(P ), formally defined below. The universal monoid Υ(P ) of Cat(P ) will be called the interval monoid of P , and it is a far more special concept than the universal monoid of an arbitrary category. By applying the results of Section 5 to that monoid, we will get a simple criterion for Υ(P ) being a gcd-monoid, always satisfied for the barycentric subdivision of a simplicial complex. This will enable us, in Theorem 7.10, to describe any group in terms of a gcd-monoid of the form Υ(P ), at least up to a doubly free factor. Definition 7.1. For a poset P , we denote by Cat(P ) the category consisting of all closed intervals [x, y], where x ≤ y in P , with (partial) multiplication given by

[x, z] = [x, y] • [y, z] , whenever x ≤ y ≤ z in P .
By identifying any x ∈ P with the singleton interval [x, x] = {x}, we obtain that the source and target map on Cat(P ) are given by ∂ 0 [x, y] = x and ∂ 1 [x, y] = y, for all x, y ∈ P with x ≤ y.

In the "object-arrow" view of Cat(P ), the objects are the elements of P , with exactly one arrow from x to y if x ≤ y, no arrow if x y. Definition 7.2. Let P be a poset. The interval monoid (resp., interval group) of P is defined as Υ(P ) = U mon (Cat(P )) (resp., Υ ± (P ) = U gp (Cat(P ))).

It follows that a system of generators and relations for both Υ(P ) (within monoids) and Υ ± (P ) (within groups) is given as follows: the generators are (indexed by) the closed intervals [x, y], where x ≤ y in P (write [x, y] P in case P needs to be specified), and the defining relations are those of the form [x, x] = 1 , whenever x ∈ P , (7.1)

[x, z] = [x, y] • [y, z] ,
whenever x ≤ y ≤ z within P . (7.2)

Remark 7.3. For a poset P and a commutative, unital ring K, the semigroup algebra K[Υ(P )] of the monoid Υ(P ) bears some similarities with the incidence algebra of P over K (the latter concept originates in Rota [START_REF] Rota | On the foundations of combinatorial theory. I. Theory of Möbius functions[END_REF]). However, the two algebras are not isomorphic as a rule: for example,

y 1 = x 2 implies that [x 1 , y 1 ]•[x 2 , y 2 ]
vanishes within the incidence algebra of P , but not within K[Υ(P )].

In particular, if K is a field and P is a finite poset with a non-trivial interval x < y, the incidence algebra of P over K is finite-dimensional, while all powers [x, y] n , for n ∈ N, are distinct in Υ(P ), so K[Υ(P )] is infinite-dimensional.

Proposition 7.4. Υ ± (P ) ∼ = Υ ± (Sim(P )), for every poset P .

Proof. The elements [x, y] Sim(P ) of Υ ± (Sim(P )), for x ≤ y in P , satisfy the defining relations (7.1) and ( 7.2) of Υ ± (P ), thus there is a unique group homomorphism ϕ : Υ ± (P ) → Υ ± (Sim(P )) such that ϕ([x, y] P ) = [x, y] Sim(P ) whenever x ≤ y in P .

A similar argument, together with an easy case inspection, shows that there is a unique group homomorphism ψ : Υ ± (Sim(P )) → Υ ± (P ) such that

ψ([x, y] Sim(P ) ) = [x, y] P , if x ≤ y , [y, x] -1 P , if y ≤ x ,
for every chain {x, y} of P .

Obviously, ϕ and ψ are mutually inverse.

By invoking Proposition 6.3, Proposition 7.4 can be used for quick calculations of Υ ± (P ) for finite P . Set π 1 (P, p) = def π 1 (Sim(P ), p), whenever p ∈ P . 7.1 and denote by K its chain complex. Since P has a lower bound (viz. p), π 1 (P, p) is trivial. This can also be verified directly by applying Tietze's Theorem to the spanning tree E represented in thick lines in Figure 7.1. Since E has 6 elements and by Propositions 6.3 and 7.4, it follows that Υ ± (P ) ∼ = F gp [START_REF]Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC[END_REF].

The following result expresses a quite special property of interval monoids of posets, not shared by all universal monoids of categories as a rule. Proposition 7.6. Let P be a poset. Then the interval monoid Υ(P ) embeds into the free group on P . In particular, the canonical map Υ(P ) → Υ ± (P ) is a monoid embedding.

Proof. Set µ(x, y) = def x -1 y, for all x ≤ y in P . Trivially, µ(x, x) = 1 and µ(x, z) = µ(x, y) • µ(y, z) whenever x ≤ y ≤ z in P . Hence, there is a unique monoid homomorphism µ : Υ(P ) → F gp (P ) such that µ([x, y]) = x -1 y , whenever x ≤ y in P .

It thus suffices to prove that µ is one-to-one. Let a be an element of Υ(P ) = U mon (Cat(P )), written in reduced form as

a = [x 1 , y 1 ] • [x 2 , y 2 ] • • • [x n , y n ] , where x i < y i whenever 1 ≤ i ≤ n . (7.3)
(cf. Section 3). The fact that the right hand side of (7.3) is a reduced finite sequence means that y i = x i+1 whenever 1 ≤ i < n. Applying the homomorphism µ to (7.3), we obtain

µ(a) = x -1 1 y 1 x -1 2 y 2 • • • x -1 n y n . (7.4)
Since y i = x i+1 whenever 1 ≤ i < n, the word written on the right hand side of (7.4) is the unique reduced word representing the element µ(a) of F gp (P ). Hence, µ(a) determines the sequence (x 1 , y 1 , x 2 , y 2 , . . . , x n , y n ), and hence it determines a, thus completing the proof that µ is one-to-one.

Proposition 7.6 will get strengthened further in Proposition 8.2, where, in the finite case, "free group" will be improved to "free monoid". Remark 7.7. It is known since Bergman [2, Example 23] that there are monoids M , embeddable into free groups, of which the universal group is not free. This remark applies to the monoids Υ(P ): although, by Proposition 7.6, the monoid Υ(P ) always embeds into a free group, it will follow from Theorem 7.10 that every group is a doubly free factor of some Υ ± (P ); and thus, Υ ± (P ) may not be free (e.g., it may have torsion).

Not every Υ(P ), for P a poset, is a gcd-monoid. Nevertheless, the posets P such that Υ(P ) is a gcd-monoid can be easily characterized. We set

P a = {x ∈ P | x ≤ a} , (7.5)
P a = {x ∈ P | x ≥ a} , (7.6)
for each a ∈ P .

Proposition 7.8. The following statements hold, for any poset P :

(1) Υ(P ) is a left gcd-monoid iff P a is a meet-semilattice for every a ∈ P .

(2) Υ(P ) is a right gcd-monoid iff P a is a join-semilattice for every a ∈ P .

Proof. By symmetry, it suffices to establish (1). It follows from Theorem 5.9 that the monoid Υ(P ) = U mon (Cat(P )) is a left gcd-monoid iff Cat(P ) is a left gcdcategory. By definition, the latter statement means that for all x, y 1 , y 2 ∈ P , if x ≤ y 1 and x ≤ y 2 , then the left gcd [x, y 1 ] ∧ Cat(P ) [x, y 2 ] is defined. Now it is straightforward to verify that the left divisibility preordering of Cat(P ) is given by

[u 1 , v 1 ] Cat(P ) [u 2 , v 2 ] ⇔ (u 1 = u 2 and v 1 ≤ v 2 ) , for all u 1 , u 2 , v 1 , v 2 ∈ P .
In particular, [x, y 1 ] ∧ Cat(P ) [x, y 2 ] is defined iff {y 1 , y 2 } has a greatest lower bound in P x . The desired conclusion follows immediately.

As an immediate consequence, observe the following.

Corollary 7.9. Let P be the barycentric subdivision of a simplicial complex K. Then Υ(P ) is a gcd-monoid.

Proof. The elements of P are, by definition (cf. Section 2), the simplices of K. For every simplex a of K, the subset P a = {x ∈ P | x ⊆ a} is a join-semilattice (the join of {x 1 , x 2 } is x 1 ∪x 2 ; it is nonempty because it contains x 1 , and it is contained in a, thus it is a simplex), and the subset

P a = {x ∈ P | a ⊆ x} is a meet-semi- lattice (the meet of {x 1 , x 2 } is x 1 ∩ x 2 ;
it is nonempty because it contains a, and it is contained in x 1 , thus it is a simplex). The desired conclusion follows from Proposition 7.8.

The following result shows that loosely speaking, groups and gcd-monoids, and even interval monoids of posets, are objects of similar complexity.

Theorem 7.10. For every group G, there exists a connected poset P such that G is a doubly free factor of Υ ± (P ) and such that the following conditions hold:

(1) P has height 2.

(2) Υ(P ) is a gcd-monoid. Furthermore, P can be taken to be finite iff G is finitely presented.

Proof. It is well known (see, for example, Rotman [START_REF]An Introduction to the Theory of Groups[END_REF]Theorem 11.64]) that G is isomorphic to the fundamental group of some connected simplicial complex K, which, in addition, is finite in case G is finitely presented.

Observe that the simplicial complex K constructed in the proof of [START_REF]An Introduction to the Theory of Groups[END_REF]Theorem 11.64] has dimension 2. (Alternatively, the fundamental group of K depends only on the 2-skeleton K (2) .)

Since K is a connected simplicial complex, the barycentric subdivision P of K (cf. Section 2) is a connected poset. Since K has dimension 2, P has height 2. Now it is well known that K and Sim(P ) have isomorphic fundamental groups: for example, K and Sim(P ) have the same geometric realizations (cf. Rotman [16, Exercise 7.12(i)]), and the fundamental group of K depends only on the geometric realization of K (cf. Rotman [START_REF] Rotman | An Introduction to Algebraic Topology[END_REF]Theorem 7.36]; see also Reynaud [14, Section 2]). By Proposition 6.3, it follows that G is a doubly free factor of Υ ± (Sim(P )). By Proposition 7.4, Υ ± (Sim(P )) = Υ ± (P ), thus G is a doubly free factor of Υ ± (P ). Moreover, it follows from Corollary 7.9 that Υ(P ) is a gcd-monoid.

Finally, if P is finite, then Υ(P ) is finitely presented, thus so is each of its retracts. Now every free factor is a retract. We will see in the next section that if P is finite (which can be ensured iff G is finitely presented), then the interval monoid Υ(P ) can be embedded into a free monoid.

Embedding gcd-monoids into free monoids

In contrast to Proposition 7.6, not every gcd-monoid can be embedded into a free monoid, or even into a free group. For example, the braid monoid B + 3 , defined by the generators a, b and the unique relation aba = bab, is a gcd-monoid (cf. Dehornoy [START_REF]Foundations of Garside Theory[END_REF]Proposition IX.1.26]), but it cannot be embedded into any free group: for example, any elements a and b in a free group such that aba = bab satisfy (a 2 b) 2 = (aba) 2 , thus, since free groups are bi-orderable (see, for example, Dehornoy et al. [START_REF] Dehornoy | Why are Braids Orderable?[END_REF]Subsection 9.2.3]), a 2 b = aba, and thus ab = ba, and hence (since aba = bab) a = b, which does not hold in B + 3 .

However, for interval monoids the situation is different. It is obvious that the assignment P → Υ(P ) (resp., P → Υ ± (P )) defines a Functor, from the Category Pos of all posets, to the Category Mon (resp., Gp) of all monoids (resp., groups). More can be said. Lemma 8.1. Let P and Q be posets and let f : P → Q be an isotone map. Then there is a unique monoid homomorphism Υ(f ) : Υ(P ) → Υ(Q) such that

Υ(f )([x, y] P ) = [f (x), f (y)] Q whenever x ≤ y in P .
Moreover, if f is one-to-one, then so is Υ(f ).

Proof. The existence and uniqueness of Υ(f ) follow trivially from the universal property defining Υ(P ). Now assume that f is one-to-one. Let a ∈ Υ(P ), written in reduced form as

a = [x 1 , y 1 ] P • [x 2 , y 2 ] P • • • [x n , y n ] P ,
where each x i < y i in P , and

y i = x i+1 whenever 1 ≤ i < n. Then Υ(f )(a) = [f (x 1 ), f (y 1 )] Q • [f (x 2 ), f (y 2 )] Q • • • [f (x n ), f (y n )] Q , (8.1)
and since f is one-to-one, the sequence written on the right hand side of (8.1) is also reduced. By the uniqueness of the reduced form of an element of Υ(Q) (cf. Lemma 3.4), it follows that Υ(f )(a) determines the finite sequence

(f (x 1 ), f (y 1 ), . . . , f (x n ), f (y n )) .
Since f is one-to-one, Υ(f )(a) also determines the finite sequence (x 1 , y 1 , . . . , x n , y n ) , thus it determines a.

The following easy result, which improves Proposition 7.6 in the finite case, is the central part of the present section. Proposition 8.2. Let P be a finite poset. Then Υ(P ) embeds into a (finitely generated ) free monoid.

Proof. It is well known that every poset P has a linear extension P , that is, a linear ordering on the underlying set of P extending the ordering of P . By applying Lemma 8.1 to the identity map f : P → P , we see that Υ(f ) is a monoid embedding Υ(P ) ֒→ Υ( P ). Now denote by x 1 , . . . , x n the elements of P listed in increasing order. It is straightforward to verify that Υ( P ) is a free monoid, with free generators [x i , x i+1 ] P where 1 ≤ i < n.

Example 8.3. Proposition 8.2 does not extend to infinite posets. For example, letting

P = {0}∪{1/n | n ∈ N \ {0}}, the sequence ([0, 1/n] | n ∈ N \ {0}
) is strictly decreasing in Υ(P ) with respect to the left divisibility ordering. Hence, Υ(P ) cannot be embedded into any free monoid.

As the following example shows, the most natural converse of Proposition 8.2, namely whether every finitely presented submonoid of a finitely generated free monoid is an interval monoid, does not hold.

Example 8.4. A finitely presented submonoid of N which is not the interval monoid of any poset.

Proof. The submonoid M = N \ {1} of the free monoid N can be defined by the generators a, b and the relations ab = ba, a 3 = b 2 (think of a and b as 2 and 3, respectively, and remember that we are dealing with the additive structure of N).

Suppose that M ∼ = Υ(P ), for a poset P . Since the elements a and b are atoms of M (i.e., none of them can be expressed as the sum of two nonzero elements of M ), each of them is a standard generator of Υ(P ), that is, a = [x, y] and b = [x ′ , y ′ ] for some x < y and x ′ < y ′ in P . Since the finite sequences ([x, y], [x, y], [x, y]) and ([x ′ , y ′ ], [x ′ , y ′ ]) are both reduced, and they represent a 3 and b 2 , respectively, we get a 3 = b 2 , a contradiction.

Remark 8.5. The argument of Example 8.4 can be easily expanded to prove that A submonoid of N is the interval monoid of a poset iff it is cyclic (i.e., of the form mN where m ∈ N).

For those variants of Example 8.4, the difference between "finitely generated" and "finitely presented" is immaterial, because every finitely generated commutative monoid is finitely presented (this is Redei's Theorem, see for example Freyd [START_REF] Freyd | Redei's finiteness theorem for commutative semigroups[END_REF]) and every submonoid of N is finitely generated.

In the non-commutative case, the situation becomes different. To begin with, a finitely generated submonoid of a free monoid may not be finitely presented (cf. Spehner [START_REF] Spehner | Présentations et présentations simplifiables d'un monoïde simplifiable[END_REF]Theorem 3.2] or [START_REF]Every finitely generated submonoid of a free monoid has a finite Mal ′ cev's presentation[END_REF]Example 2.9]). Since interval monoids of finite posets are finitely presented, let us focus attention on finitely presented monoids.

Example 8.6. A finitely presented submonoid of a free monoid, which is also a gcd-monoid, but which is not isomorphic to the interval monoid of any poset. (originating in Dehornoy [START_REF] Dehornoy | Complete positive group presentations[END_REF]). The presentation (8.2) is right complemented, which means that for every pair (s, t) of generators, there is at most one relation of the form sx = ty in the presentation. Moreover, the right complements x and y solving this problem are themselves generators. This makes it very easy to verify the right cube condition (cf. [START_REF]Foundations of Garside Theory[END_REF]Subsection II.4.4]) on that presentation. It thus follows from the results of [START_REF]Foundations of Garside Theory[END_REF]Section II.4] that M 6 is a left gcd-monoid. A symmetric argument yields that M is a right gcd-monoid.

Further, in order to verify the right 3-Ore condition introduced in Dehornoy [START_REF]Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC[END_REF], namely that any three elements of M 6 , which pairwise admit a right multiple, admit a common right multiple, it suffices (cf. [START_REF]Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC[END_REF]Section 5]) to verify that condition on any subset X of M 6 containing the atoms and closed under right complementation; in the present case, just take X = Σ. The left 3-Ore condition is verified similarly. By [START_REF]Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC[END_REF]Section 4], it follows that M 6 can be embedded into its universal group G = U gp (M 6 ). (The latter fact is also a straightforward application of Adjan's Theorem [START_REF] Sergeȋ | Defining relations and algorithmic problems for groups and semigroups[END_REF], see also Remmers [13,Theorem 4.6].)

Now denote by x and y two new generators and set Ω = {a, b, x, y}. Denote by M ′ 6 the submonoid of F mon (Ω) generated by Σ

′ = {a ′ , b ′ , c ′ , d ′ , e ′ , f ′ } where a ′ = a, b ′ = b, c ′ = ax, d ′ = by, e ′ = xb, f ′ = ya. Since a ′ e ′ = c ′ b ′ and d ′ a ′ = b ′ f ′ , there is a unique monoid homomorphism ε : M 6 → M ′
6 sending each generator of Σ to its primed version (i.e., ε(a) = a ′ , and so on).

The following claim shows that M 6 embeds into the free monoid F mon (Ω).

Claim. The map ε is an isomorphism from M 6 onto M ′ 6 . Proof of Claim. Within the group G = U gp (M 6 ), we set x = a -1 c = eb -1 and y = b -1 d = f a -1 . We denote by ϕ : F mon (Ω) → G the unique monoid homomorphism such that ϕ(a) = a, ϕ(b) = b, ϕ(x) = x, and ϕ(y) = y, and we denote by η the restriction of ϕ to M ′ 6 . Then η(a ′ ) = a, η(b ′ ) = b, η(c ′ ) = ax = c, η(d ′ ) = by = d, η(e ′ ) = xb = e, and η(f ′ ) = ya = f , whence η • ε is the inclusion map M 6 ֒→ G. In particular, ε is one-to-one. But ε is, by construction, surjective.

Claim.

Therefore, in order to conclude the proof of Example 8.6, it suffices to prove that M 6 is not isomorphic to the interval monoid of any poset. Let M 6 ∼ = Υ(P ), for a poset P . Since the elements of Σ are all atoms of M 6 , they are all standard generators of Υ(P ). Since ae = cb and a = c, none of the finite sequences (a, e) and (c, b) is reduced, thus there are elements o, p, q, i ∈ P such that o < p < i and o < q < i within P , and a

= [o, e = [p, i], c = [o, q], b = [q, i].
In particular, min a = o < q = min b. A similar argument, applied to the relation bf = da, yields the relation min b < min a; a contradiction.

Gcd-monoids arising from extreme spindles

In this section we introduce a class of categories, associated to certain intervals in posets, closely related to the categories Cat(P ). Those categories are a source of counterexamples in the paper Dehornoy and Wehrung [START_REF] Dehornoy | Multifraction reduction III: The case of interval monoids[END_REF]; in particular, as we will see in that paper, the universal monoid of such a category, although cancellative, may not embed into any group.

Those categories (cf. Definition 9.3) are built from the following concept.

u, u] • X = X • [v, v] = X , whenever X ∈ C u,v , (9.1)
[x, y] • [y, z] = [x, z] , whenever x ≤ y ≤ z and (x, z) = (u, v) , (9.2)
[u, z] • [z, v] = Z , whenever u < z < v , Z ∈ C u,v , and z ∈ Z . (9.
3)

The verification of the following technical lemma is tedious, but straightforward, and we omit its proof. Lemma 9.4. Let [u, v] be an extreme spindle in a poset P . Then Cat(P, u, v), endowed with the multiplication given in Definition 9.3, is a category. Furthermore, the divisibility orderings and , on that category, are given by

[x, y] [x ′ , y ′ ] iff x = x ′ and y ≤ y ′ , whenever [x, y], [x ′ , y ′ ] = [u, v] , Z [x, y] , whenever [x, y] = [u, v] and Z ∈ C u,v , [x, y] Z iff x = u and y ∈ Z , whenever [x, y] = [u, v] and Z ∈ C u,v , X Y iff X = Y , whenever X, Y ∈ C u,v , and
[x, y] [x ′ , y ′ ] iff x ′ ≤ x and y = y ′ , whenever [x, y], [x ′ , y ′ ] = [u, v] , Z [x, y] , whenever [x, y] = [u, v] and Z ∈ C u,v , [x, y] Z iff y = v and x ∈ Z , whenever [x, y] = [u, v] and Z ∈ C u,v , X Y iff X = Y , whenever X, Y ∈ C u,v .
By identifying any x ∈ P with the singleton interval [x, x] = {x}, we obtain that the source and target map, on Cat(P, u, v), are given by

∂ 0 [x, y] = x and ∂ 1 [x, y] = y , whenever [x, y] = [u, v] , ∂ 0 Z = u and ∂ 1 Z = v , whenever Z ∈ C u,v .
Proposition 9.5. Let P be a poset such that Υ(P ) is a gcd-monoid, and let [u, v] be an extreme spindle of P . Then Cat(P, u, v) is a gcd-category, and its universal monoid Υ(P, u, v) = def U mon (Cat(P, u, v)) is a gcd-monoid.

Proof. It follows from Proposition 7.8 that P a is a join-semilattice and P a is a meet-semilattice, for every a ∈ P . Then a direct application of Lemma 9.4 yields that Cat(P, u, v) is a gcd-category, where the left gcd (resp., right gcd), of a pair of elements with the same source (resp., target), are respectively given by

[x, y] ∧ [x, z] = [x, y ∧ z] , for all [x, y], [x, z] = [u, v] , [u, x] ∧ Z = [u, max([u, x] ∩ Z)] , whenever u ≤ x , x = v , and Z ∈ C u,v , [x, z] ∧ [y, z] = [x ∨ y, z] , for all [x, z], [y, z] = [u, v] , [x, v] ∧ Z = [min([x, v] ∩ Z), v] , whenever x ≤ v , x = u , and Z ∈ C u,v , X ∧ Y = X ∧ Y = {u} , for all X = Y in C u,v .
By Theorem 5.9, it follows that Υ(P, u, v) is a gcd-monoid.

It is easy to construct examples showing that the assumption of Proposition 9.5, which is equivalent to saying that each P a is a join-semilattice and each P a is a meet-semilattice, can be relaxed. For example, defining P by its covering relations 0 < p i and p i < q j , whenever i, j ∈ {0, 1}, with the extreme spindle [0, q 0 ], it is easy to verify that Cat(P, 0, q 0 ) is a gcd-category. Nevertheless, P 0 = P is not a meet-semilattice, because {q 0 , q 1 } has no greatest lower bound. Proposition 9.6. Let P be a poset and let [u, v] be an extreme spindle of P . Then the monoid Υ(P, u, v) can be defined by the generators [x, y], where x < y in P with (x, y) = (u, v), and the relations [x, z] = [x, y] • [y, z] , whenever x < y < z in P with (x, z) = (u, v) .

(9.4)

Proof. We verify that M 0 = def Υ(P, u, v) satisfies the universal property defining the monoid presented by the relations (9.4). Thus let M be a monoid, with elements a x,y ∈ M , for (x, y) = [u, v], such that a x,z = a x,y a y,z whenever x < y < z and (x, z) = (u, v). We need to prove that there is a unique monoid homomorphism ϕ : M 0 → M such that ϕ([x, y]) = a x,y whenever x < y and (x, y) = (u, v).

We first extend the function (x, y) → a x,y by setting a x,x = 1 whenever x ∈ P . For the existence part, it suffices to prove that M 0 is generated by the subset Cat(P ) \ {[u, v]}. Let Z be a maximal chain of [u, v]. Since [u, v] = {u, v}, there is z ∈ Z such that u < z < v. Hence, within Cat(P, u, v) (thus within M 0 ), Z = [u, z] • [z, v], as required. This completes the proof of the uniqueness part.

Let us deal with existence now. We claim that for any maximal chain Z of [u, v], the element a u,z a z,v , where z ∈ Z \ {u, v}, is independent of z. Indeed, any x, y ∈ Z \ {u, v} are comparable, say x ≤ y, and then a u,x a x,v = a u,x a x,y a y,v = a u,y a y,v , thus proving our claim. Denote by b Z the common value of all a u,z a z,v , where z ∈ Z \ {u, v}. We need to prove that the a x,y , where [x, y] = [u, v], and the b Z , where Z ∈ C u,v , satisfy the relations defining the monoid M 0 , which are also the relations (9.1)-(9.3) defining the category Cat(P, u, v). The only non-trivial instances to be verified are a x,z = a x,y a y,z whenever x ≤ y ≤ z and (x, z) = (u, v), and b Z = a u,z a z,v whenever Z ∈ C u,v and z ∈ Z \ {u, v}, which all hold by construction.

A criterion of group-embeddability for universal monoids

For any category S, the universal property defining U mon (S) implies immediately that there is a unique monoid homomorphism ϕ S : U mon (S) → U gp (S) such that ϕ S • ε S = η S . In fact, ϕ S = η Umon(S) . Since there are monoids (even cancellative ones) that cannot be embedded into groups (see Meakin [START_REF] Meakin | Groups and semigroups: connections and contrasts, Groups St[END_REF] for a survey), ϕ S may not be an embedding. However, the result below states a convenient criterion for this to occur. It states that for any category S, the embeddability of U mon (S) into a group can be verified "locally", that is, on the hom-sets. The main trick used in the proof of Theorem 10.1 will be called the highlighting expansion of a morphism.

Theorem 10.1. The following are equivalent, for any category S:

(i) The canonical map ϕ S : U mon (S) → U gp (S) is one-to-one.

(ii) U mon (S) embeds into a group.

(iii) There are a group G and a functor ψ : S → G such that the restriction of ψ to every hom-set of S is one-to-one.

Proof. (i)⇒(ii) is trivial, while (ii)⇒(iii) follows immediately from Lemma 3.10.

Finally let us prove (iii)⇒(i). Let G and ψ be as prescribed in (iii). We need to prove that ϕ S is one-to-one. Our proof will be an amplification of the one of Proposition 7.6.

The map ψ ′ : S → F gp (Idt S) * G, x → (∂ 0 x) -1 ψ(x)∂ 1 x is a functor. It works by adding to ψ(x) the endpoints information on x, thus we will call it the highlighting expansion of ψ.

Claim. The restriction of ψ ′ to every hom-set of S is one-to-one. Furthermore, ψ ′ (x) = 1 implies that x ∈ Idt S, for any x ∈ S.

Proof of Claim. Let a, b ∈ Idt S and let x, y ∈ S(a, b) such that ψ ′ (x) = ψ ′ (y). This means that a -1 ψ(x)b = a -1 ψ(y)b within F gp (Idt S) * G, that is, ψ(x) = ψ(y), so, by assumption, x = y. Now let x ∈ S such that ψ ′ (x) = 1. Setting a = ∂ 0 x and b = ∂ 1 x, this means that a -1 ψ(x)b = 1 within F gp (Idt S) * G, so, by the uniqueness of the normal form for elements of that group (see, for example, Rotman [START_REF]An Introduction to the Theory of Groups[END_REF]Theorem 11.52]), a = b and ψ(x) = 1, and so ψ(x) = ψ(a), where x and a both belong to S(a, a). By our assumption, it follows that x = a.

Claim.

By the Claim above, we may replace (G, ψ) by its highlighting expansion F gp (Idt S) * G, ψ ′ and thus assume from the start that ψ(x) = 1 implies that x ∈ Idt S , for each x ∈ S .

(10.1)

Forming again the highlighting expansion ψ ′ of that new map ψ, it follows from the universal property defining U gp (S) that there is a unique group homomorphism σ : U gp (S) → F gp (Idt S) * G such that ψ ′ = σ • η S . Now let x, y ∈ U mon (S) such that ϕ S (x) = ϕ S (y). Writing x red = (x 1 , . . . , x m ) and y red = (y 1 , . . . , y n ), this can be written Since each b i = a i+1 (because x i x i+1 ↑) and c j = d j+1 (because y j y j+1 ↑), and since, by (10.1), each ψ(x i ) and each ψ(y j ) belongs to G \ {1}, it follows from the uniqueness of the normal form for elements of F gp (Idt S) * G that m = n, each (a i , b i ) = (c i , d i ), and each ψ(x i ) = ψ(y i ). Since x i and y i both belong to S(a i , b i ), it follows from our assumption that x i = y i . Therefore, x = y.

By defining ψ as the constant map with value 1, we get immediately part of the result of Proposition 7.6 -namely, that the interval monoid of a poset always embeds into a group. As the following example shows, the range of application of Theorem 10.1 goes beyond universal monoids of ordered sets.

Lemma 3 . 3 .( 1 ) 2 )

 3312 The following statements hold: The union of the binary relation →, with the equality, is confluent: that is, whenever a → b i , for each i ∈ {0, 1}, then either b 0 = b 1 or there exists c ∈ Seq S such that b 0 → c and b 1 → c. (The binary relation → * is also confluent.

Proposition 4 . 1 .

 41 A category S is conical iff its universal monoid U mon (S) is conical.Proof. Let U mon (S) be conical and let a, b ∈ S such that ab is an identity. Then ε S (a)ε S (b) = ε S (ab) = 1, thus, as U mon (S) is conical, ε S (a) = ε S (b) = 1, and thus a and b are both identities.

. 1 )

 1 Now let x = (x 1 , . . . , x m ) and y = (y 1 , . . . , y n ) be elements in U mon (S) (reduced sequences) such that ε S (a)x = ε S (a)y; we must prove that x = y. We may assume that m ≤ n. It follows from (4.1) that either n = m or n = m + 1. Since the case where m = n = 0 is trivial, there are three cases left to consider.Case 1. m = n > 0 and ax 1 , ay 1 are both undefined. Then (a, x 1 , . . . , x m ) = ε S (a)x = ε S (a)y = (a, y 1 , . . . , y m ) ,thus x i = y i whenever 1 ≤ i ≤ m.Case 2. m = n > 0 and ax 1 , ay 1 are both defined. Then (ax 1 , x 2 , . . . , x m ) = ε S (a)x = ε S (a)y = (ay 1 , y 2 , . . . , y m ) , thus ax 1 = ay 1 and x i = y i whenever 2 ≤ i ≤ m. Furthermore, from the first equation and the left cancellativity of a it follows that x 1 = y 1 .Case 3. n = m + 1, ax 1 is undefined, and ay 1 is defined. Then (a, x 1 , . . . , x m ) = ε S (a)x = ε S (a)y = (ay 1 , y 2 , . . . , y m+1 ) , thus ay 1 = a = a∂ 1 a, and thus y 1 = ∂ 1 a, an identity of S; in contradiction with the sequence y being reduced.

  Proof. It is sufficient to prove the statements about . The equivalence between ε S (a) M 1 and a S ∂ 0 a follows from Lemma 4.2(3). Now let a S ∂ 0 a and b ∈ M \ {1}, we need to prove that ε S (a) M b iff a S ∇ 0 b. The implication from the right to the left is obvious since then, ε S (a) M ε S (∇ 0 b) M b. Suppose, conversely, that ε S (a) M b, we need to prove that a S ∇ 0 b. Let

For 1 c→ e 2

 12 an interpretation of Lemma 5.3 in terms of normal forms and Garside families, see Remark 5.11. Lemma 5.4. Let S be a conical category. If S is left (resp., right ) cancellative, then the left (resp., right ) divisibility preordering of S is a partial ordering. Proof. We prove the statement about the left divisibility preordering S of S. Let a S b S a, where a, b ∈ S. There are x, y ∈ S such that b = ax and a = by; whence a∂ 1a = a = axy. Since S is left cancellative, ∂ 1 a = xy. Since S is conical, x = y = ∂ 1 a, so a = b.Although the conicality assumption cannot be weakened in Lemma 5.4, the cancellativity assumption can. For example, defining S as the category with three objects e 0 , e 1 , e 2 and arrows a, b with e 0 a,b → e 1 and e with ac = bc, both left and right divisibility preorderings of S are orderings, while S is not left cancellative.Say that a finite sequence a = (a 1 , . . . , a m ) is a prefix (resp., suffix) of a finite sequence b = (b 1 , . . . , b n ) if m ≤ n and a i = b i (resp., a i = b n-m+i ) whenever 1 ≤ i ≤ m.It is trivial that Umon(S) (resp., Umon(S) ) extends the prefix ordering (resp., suffix ordering) on U mon (S). As the following lemma shows, more is true. Lemma 5.5. Let S be a conical category and set M = U mon (S). The following statements hold, for all a, b ∈ U mon (S): (1) a M b iff either a is a prefix of b or there are a ′ , b ′ ∈ U mon (S) and u, v ∈ S such that a = a ′ (u), b = a ′ (v) b ′ , and u S v.

( 2 )

 2 a M b iff either a is a suffix of b or there are a ′ , b ′ ∈ U mon (S) and u, v ∈ S such that a = (u) a ′ , b = b ′ (v) a ′ , and u S v. Proof. By symmetry, it is sufficient to prove (1). Write a = (a 1 , . . . , a m ) and b = (b 1 , . . . , b n ).Suppose that a M b and let x ∈ M such that b = ax. We establish the condition stated in the right hand side of (1). Write x = (x 1 , . . . , x k ) (a reduced sequence). It follows from Corollary 3.8 that m ≤ n. Furthermore, by Lemma 3.7, there are two cases to consider.Case 1. a m x 1 is undefined. Then b = a x, so a is a prefix of b and the desired conclusion holds.Case 2. a m x 1 is defined. The desired conclusion holds with a ′ = (a 1 , . . . , a m-1 ), b ′ = (x 2 , . . . , x k ), u = a m , and v = a m x 1 .

( 1 ) 2 )

 12 c M a i for all i ∈ I iff either c M a left , or a left = a i for each i ∈ I and there is c ∈ S such that c = a left (c) and c S ∇ 0 (a i M a left ) for each i ∈ I. (Suppose that S is left cancellative. Then c is the left gcd of {a i | i ∈ I} in M iff one of the following cases occurs: (i) c = a left , and either a left = a i for some i, or {∂ 0 ∇ 0 (a i M a left ) | i ∈ I} is not a singleton, or the left gcd of {∇ 0 (a i M a left ) | i ∈ I} is an identity; (ii) a i = a left for all i, the left gcd c of {∇ 0 (a i M a left ) | i ∈ I} exists in S and it is not an identity, and c = a left (c). (3) c M a i for all i ∈ I iff either c M a right , or a right = a i for each i ∈ I and there is c ∈ S such that c = (c) a right and c S ∇ 1 (a i M a right ) for each i ∈ I. (4) Suppose that S is right cancellative. Then c is the right gcd of {a i | i ∈ I} in M iff one of the following cases occurs:

Corollary 5 . 7 .

 57 or the only left divisor of the set {∇ 0 (a i M a left ) | i ∈ I} is an identity. Again by (1), the gcd of {a i | i ∈ I} exists and is not equal to a left iff it has the form a left (c) where c is the left gcd of {∇ 0 (a i M a left ) | i ∈ I} in S and c is not an identity. A direct application of Lemma 5.6 to one-entry sequences yields the following. The following statements hold, for any conical category S and any a, b ∈ S: (1) If S is left cancellative and ∂ 0 a = ∂ 0 b, then ε S (a) ∧ ε S (b) exists in U mon (S) iff a ∧ b exists in S. Furthermore, if this holds, then ε S (a ∧ b) = ε S (a) ∧ ε S (b). (2) If S is right cancellative and ∂ 1 a = ∂ 1 b, then ε S (a) ∧ ε S (b) exists in U mon (S) iff a ∧ b exists in S. Furthermore, if this holds, then ε S (a ∧ b) = ε S (a) ∧ ε S (b). Definition 5.8. A category S is a left gcd-category if it is conical, left cancellative, and any elements a, b ∈ S such that ∂ 0 a = ∂ 0 b have a left gcd. Right gcd-categories are defined dually. We say that S is a gcd-category if it is simultaneously a left and right gcd-category.

Proof.

  It suffices to establish the statement for left gcd-categories and left gcdmonoids. Set M = U mon (S). It follows from Proposition 4.1 and Corollary 4.3 that S is conical and left cancellative iff M is conical and left cancellative. Now suppose that M is a left gcd-monoid and let a, b ∈ S such that ∂ 0 a = ∂ 0 b. We must prove that {a, b} has a left gcd in S. We may assume that a = b and neither a nor b is an identity. Hence (a) and (b) are both reduced sequences. By assumption, they have a left gcd in M . Since they are distinct, their largest common prefix is the empty sequence, thus, as ∇ 0 (a) = a and ∇ 0 (b) = b and by Lemma 5.6(2), a ∧ S b exists.

Proposition 5 . 10 .

 510 The following statements hold, for any gcd-category S and any a, b ∈ S: (1) If ∂ 0 a = ∂ 0 b, then ε S (a) ∨ ε S (b) exists in U mon (S) iff ε S (a) and ε S (b) have a common right multiple in U mon (S), iff a ∨ b exists in S, iff a and b have a common right multiple in S. Furthermore, if this holds, then ε S (a ∨ b) = ε S (a) ∨ ε S (b). (2) If ∂ 1 a = ∂ 1 b, then ε S (a) ∨ ε S (b) exists in U mon (S) iff ε S (a) and ε S (b) have a common left multiple in U mon (S), iff a ∨b exists in S, iff a and b have a common left multiple in S. Furthermore, if this holds, then ε S (a ∨ b) = ε S (a) ∨ ε S (b). Proof. It suffices to prove (1). The case where either a or b is an identity is trivial, so we may assume that neither a nor b are identities of S. A preliminary observation is that ε S (a) ∨ ε S (b) exists in U mon (S) iff ε S (a) and ε S (b) have a common right multiple in U mon (S), and, similarly, a ∨ b exists in S iff a and b have a common right multiple in S. This follows from the argument of Dehornoy [6, Lemma 2.15]. Now suppose that a ∨ b exists in S. Clearly, ε S (a ∨ b) is a common right multiple of ε S (a) and ε S (b). Let c be a common right multiple of ε S (a) and ε S (b). It follows from Lemma 5.3 that a and b are both left divisors of ∇ 0 c; whence a ∨ b exists in S, and is a left divisor of ∇ 0 c. Again by Lemma 5.3, it follows that ε S (a ∨ b) is a left divisor of c. Hence, ε S (a) ∨ ε S (b) exists in U mon (S), and is equal to ε S (a ∨ b). Suppose, conversely, that ε S (a) ∨ ε S (b) exists in U mon (S); denote this element by c. Since ε S (a) and ε S (b) are both left divisors of c, it follows from Lemma 5.3 that a and b are both left divisors of ∇ 0 c; whence a ∨ b exists in S.

Figure 6 . 1 .

 61 Figure 6.1. A simplicial complex of dimension 1 of K. Hence, Υ ± (K) ∼ = F gp (4).A spanning tree of K is represented with thick it has three edges. By using Tietze's Theorem, we get π 1 (K, -) ∼ = Z (this is trivial in more than one way: for example, the geometric realization of K is a circle). We get again Υ± (K) ∼ = Z * F gp (3) ∼ = F gp (4).See also Example 7.5.

3 Figure 7 . 1 .

 371 Figure 7.1. A spanning tree in a poset Example 7.5. Denote by P the poset represented in Figure7.1 and denote by K its chain complex. Since P has a lower bound (viz. p), π 1 (P, p) is trivial. This can also be verified directly by applying Tietze's Theorem to the spanning tree E represented in thick lines in Figure7.1. Since E has 6 elements and by Propositions 6.3 and 7.4, it follows that Υ ± (P ) ∼ = F gp[START_REF]Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC[END_REF].

Proof.

  Denote by M 6 the monoid defined by the set of generators Σ = {a, b, c, d, e, f } and the relations ae = cb , da = bf . (8.2) We shall analyze the monoid M 6 by using the tools of Dehornoy [5, Section II.4]

Definition 9 . 1 .Proposition 9 . 2 .

 9192 Let P be a poset and let u < v in P , with [u, v] of height ≥ 2 (i.e., u < z < v for some z). We say that the closed interval[u, v] is a spindle of P if the comparability relation on the open interval ]u, v[ = def {x ∈ P | u < x < v} is an equivalence relation. If, in addition, u is a minimal element of P and v is a maximal element of P , we say that [u, v] is an extreme spindle of P . In what follows, we shall denote by C u,v the set of all maximal chains of [u, v]. The following are equivalent, for any closed interval [u, v], of height ≥ 2, in a poset P : (i) [u, v] is a spindle of P . (ii) Any two distinct maximal chains of [u, v] meet in {u, v}. Proof. Denote by ∼ the comparability relation on ]u, v[. (i)⇒(ii). Assume (i) and set Σ = ]u, v[ /∼. Every element of Σ is a chain and ]u, v[ is the disjoint union of Σ. It follows that [u, v] = (X ∪ {u, v} | X ∈ Σ). The sets X ∪ {u, v}, where X ∈ Σ, are exactly the maximal chains of [u, v], and they meet pairwise at {u, v}. (ii)⇒(i). By assumption, the elements of C u,v are pairwise meeting at {u, v}, thus every x ∈ ]u, v[ belongs to a unique C(x) ∈ C u,v . It follows that x ∼ y iff C(x) = C(y), for all x, y ∈ ]u, v[. The desired conclusion follows easily. Definition 9.3. For any extreme spindle [u, v] in a poset P , we endow the set Cat(P, u, v) = def Cat(P ) \ {[u, v]} ∪ C u,v , with the (partial) multiplication given by [

η S (x 1 )

 1 • • • η S (x m ) = η S (y 1 ) • • • η S (y n ) within G .By applying the homomorphism σ, we getψ ′ (x 1 ) • • • ψ ′ (x m ) = ψ ′ (y 1 ) • • • ψ ′ (y n ) within F gp (Idt S) * G , which, writing (a i , b i ) = (∂ 0 x i , ∂ 1 x i ) and (c j , d j ) = (∂ 0 y j , ∂ 1 y j ), means that a -1 1 ψ(x 1 )b 1 a -1 2 ψ(x 2 )b 2 • • • a -1 m ψ(x m )b m = c -1 1 ψ(y 1 )d 1 c -1 2 ψ(y 2 )d 2 • • • c -1 n ψ(y n )d n within F gp (Idt S) * G .

Example 10 . 2 .

 102 Denote by C 6 the monoid defined by the generators a, b, c, a ′ , b ′ , c ′ and the relations ab ′ = ba ′ , bc ′ = cb ′ , ac ′ = ca ′ . (10.2)

  .1) Definition 3.6. A category S is conical if xy ∈ Idt S (i.e., xy is defined and belongs to Idt S) implies that x ∈ Idt S (equivalently, y ∈ Idt S), for any x, y ∈ S. (Of course, in that case, x = y = xy.)

We prove, with the help of Theorem 10.1, that C 6 can be embedded into a group.

We consider distinct symbols 0, 1, 2, a, b, c, a ′ , b ′ , c ′ , a, b, c and we define the category S, with objects 0, 1, 2 and nonempty hom-sets defined by S(i, i) = {i} whenever i ∈ {0, 1, 2}, S(0, 1) = {a, b, c}, S(1, 2) = {a ′ , b ′ , c ′ }, S(0, 2) = aa ′ , bb ′ , cc ′ , a, b, c , and composition defined by

By definition, C 6 = U mon (S). It is straightforward to verify that S is a gcd-category. Hence, by Theorem 5.9, C 6 is a gcd-monoid. Denote by ψ : S → Z 3 the unique functor such that ψ(a) = ψ(a ′ ) = (1, 0, 0),

. It is straightforward to verify that ψ is one-to-one on every hom-set of S. For example, the elements ψ(aa ′ ), ψ(bb ′ ), ψ(cc ′ ), ψ(a), ψ(b), ψ(c) are all distinct. Since S(i, i) is a singleton for each i ∈ {0, 1, 2}, it follows from Theorem 10.1 that C 6 embeds into a group.

It is interesting to analyze C 6 via the methods of Dehornoy [START_REF]Foundations of Garside Theory[END_REF]Section II.4]. The presentation (10.2) is both left and right complemented, and it satisfies the left and right cube conditions, which yields another proof that C 6 is a gcd-monoid. On the other hand, the right 3-Ore condition fails in C 6 , for a, b, c pairwise admit common right-multiples, but they admit no global common right-multiple. Hence the methods of [5, Section II.4] are a priori not sufficient to infer the result, established above, that C 6 embeds into its universal group.

Then comes one more surprise. Eliminating, in that order, the variables c ′ and b ′ from (10.2) yields c ′ = b -1 cb ′ , then b ′ = c -1 ba -1 ca ′ , and then ac -1 b = bc -1 a. Therefore, the universal group G 6 of C 6 can be defined by generators a, b, c, a ′ and the unique relation ac -1 b = bc -1 a. Changing c to c -1 , we obtain the alternate presentation of G 6 , with generators a, b, c, a ′ , and relation acb = bca. Hence, G 6 is also the universal group of the monoid D 4 defined by generators a, b, c, a ′ , and relation acb = bca. By applying to that presentation the methods of Dehornoy [START_REF]Foundations of Garside Theory[END_REF]Section II.4], it can be verified that D 4 is a noetherian gcd-monoid satisfying both left and right 3-Ore conditions. By applying Dehornoy [6, Section 4], this gives another proof that D 4 embeds into its universal group. However, the embeddability of D 4 into its universal group would not be sufficient, a priori, to infer the above result that C 6 embeds into its universal group, because the change of presentation described above involves changing c to c -1 , which requires an ambient group.


Amirhossein Mostajabi

Mohammad Azadifar

Farhad Rachidi

Marcos Rubinstein

Gerhard Diendorfer

Wolfgang Schulz

Hannes Pichler

Vladimir A Rakov

Mario Paolone

Davide Pavanello

Simultaneous Records of Current and 380-km Distant Electric Field of a Bipolar Lightning Flash

Keywords: Bipolar lightning flash, simultaneous current and field measurement, ionospheric reflection, field-to-current conversion factor I

published or not. The documents may come

INTRODUCTION

In a bipolar flash, the current waveform exhibits a polarity reversal, corresponding to a charge transfer to ground of both negative and positive polarities [START_REF] Azadifar | An update on the charaterictics of positive flashes recorded on the Säntis Tower[END_REF], [START_REF] Romero | Some characteristics of positive and bipolar lightning flashes recorded on the Säntis tower in 2010 and 2011[END_REF]. Bipolar flashes are usually initiated by upward leaders from tall structures, but they can also be downward flashes [START_REF] Azadifar | An Update on the Instrumentation of the Säntis Tower in Switzerland for Lightning Current Measurements and Obtained Results[END_REF]. About 3% of the flashes recorded at the Säntis Tower are bipolar [START_REF] Azadifar | Bipolar lightning flashes observed at the Säntis Tower: Do we need to modify the traditional classification?[END_REF]. Fig. 1 shows the monthly distribution of negative, positive and bipolar lightning flashes accumulated over the period ranging from June 2010 to July 2016. It can be seen that positive and bipolar flashes occurred mostly during the warmer seasons (May to October).

Current waveforms associated with bipolar flashes were first reported by McEachron from his studies at the Empire State Building [START_REF] Mceachron | Lightning to the empire state building[END_REF]. Further studies on bipolar lightning are based either on lightning current observations (e.g., [START_REF] Narita | Bipolar lightning in winter at Maki, Japan[END_REF], [START_REF] Schulz | Bipolar flashes detected with lightning location systems and measured on an instrumented tower[END_REF], [START_REF] Azadifar | Bipolar lightning flashes observed at the Säntis Tower: Do we need to modify the traditional classification?[END_REF]), or electromagnetic fields, sometimes with high-speed video observations (e.g., [START_REF] Miki | Observation of upward lightning in winter at the coast of Japan Sea with a high-speed video camera[END_REF]- [START_REF] Zhou | Characteristics of upward bipolar lightning flashes observed at the Gaisberg Tower[END_REF]). Only rarely have observations been reported that include simultaneous records of currents and fields (e.g., [START_REF] Heidler | Statistics of lightning current parameters and related nearby magnetic fields measured at the Peissenberg tower[END_REF]- [START_REF] Wang | Characteristics of upward bipolar lightning derived from simultaneous recording of electric current and electric field change[END_REF]). In this paper, we present and discuss simultaneous records of current and wideband electric field waveforms at a 380 km distance from the strike point associated with an upward bipolar flash initiated from the Säntis Tower. The flash was recorded on September 21, 2014 and it contained 23 negative strokes and one positive pulse superimposed on the initial-stage current and belongs to the category I bipolar flashes according to the classification proposed by Rakov [START_REF] Rakov | A Review of Positive and Bipolar Lightning Discharges[END_REF]. In this paper, we will treat the positive pulse as a stroke.

The rest of this paper is organized as follows. Section II briefly reviews the instrumentation installed at the Säntis Tower, Switzerland and the electric field measurement station in Neudorf, Austria. A brief description of the obtained data is presented in Section III. The analysis and discussion of the results are given in Section IV. Finally, conclusions are presented in Section V.

II. EXPERIMENTAL SETUP AND INSTRUMENTATION

A. Current Measurement System at the Säntis Tower

The 124-m tall Säntis Tower has been instrumented since May 2010 using advanced equipment including remote monitoring and control capabilities for accurate measurement of lightning current parameters enabling a high-resolution sampling of lightning currents over long observation windows [START_REF] Romero | A system for the measurements of lightning currents at the Säntis Tower[END_REF], [START_REF] Azadifar | An Update on the Instrumentation of the Säntis Tower in Switzerland for Lightning Current Measurements and Obtained Results[END_REF]. The analog outputs of the sensors are relayed to a digitizing system by means of optical fiber links. The system allows over-the-Internet remote maintenance, monitoring, and control.

A PXI platform with a current sampling rate of 50MS/s is used to digitize and record measured waveforms. The lightning current is recorded over a 2.4-s time with a pretrigger delay of 960 ms.

In 2013-2014, a certain number of updates were made to the overall measuring system. More details on the measurement sensors and instrumentation system can be found in [START_REF] Azadifar | An Update on the Instrumentation of the Säntis Tower in Switzerland for Lightning Current Measurements and Obtained Results[END_REF], [START_REF] Romero | A system for the measurements of lightning currents at the Säntis Tower[END_REF]- [START_REF] Romero | Positive Lightning Flashes Recorded on the Säntis Tower in 2010 and 2011[END_REF].

B. Electric Field Measurement System

Vertical electric fields were measured at a distance of 380 km from the tower (in Neudorf, Northern Austria) by way of a flat-plate antenna and digitized with a sampling rate of 5 MS/s [START_REF] Pichler | Some Parameters of Correlated Current and Radiated Field Pulses from Lightning to the Gaisberg Tower[END_REF]. The antenna was installed on the roof of a building with an estimated enhancement factor of about 2.5 [START_REF] Pichler | Some Parameters of Correlated Current and Radiated Field Pulses from Lightning to the Gaisberg Tower[END_REF]. The measured field values were corrected to account for this enhancement. The integrator decay time constant was 0.5 ms, corresponding to a lower cutoff frequency of about 300 Hz. The triggering signal was relayed to the field measurement station using TCP/IP over the Internet from the Säntis tower. The locations of the Säntis Tower and the field measuring station in Neudorf are shown in Fig. 2. More information on the electric field measurement system in Neudorf can be found in [START_REF] Pichler | Some Parameters of Correlated Current and Radiated Field Pulses from Lightning to the Gaisberg Tower[END_REF].

III. OBTAINED DATA

We present here an upward bipolar lightning flash recorded on September 21, 2014 at 15:14:09 (local time) along with its associated vertical electric fields in Neudorf, at 380 km from the Säntis Tower. Fig. 3a shows the overall current waveform which contains an Initial Continuous Current (ICC) of negative polarity which lasted for about 320 ms (from 880 to 1200 ms) and included more than 20 superimposed pulses with peak values of about 0.6 to 6.1 kA. After this initial current of negative polarity, the flash exhibits a polarity reversal of the current during its initial stage, therefore belonging to category I according to the classification proposed by Rakov [START_REF] Rakov | A Review of Positive and Bipolar Lightning Discharges[END_REF]. It is worth noting that this bipolar flash can also be classified as being of Category III which involves return strokes of opposite polarity, assuming that the positive pulse can be treated as a stroke (which we do in this paper). The positive stroke is characterized by a peak current of 39.6 kA and a risetime of 31 µs. An expanded view of the initial stage of the current in which the ICC is resolved is shown in the inset of the figure.

The associated vertical electric field of this flash is shown in Fig. 3b. It should be noted that our field measuring system exhibited a high noise level at frequencies of 200, 300, 400 and 500 kHz. In order to reduce the noise, 20-kHz bandwidth notch filters centered at each of the above-mentioned frequencies were applied to all recorded signals.

IV. ANALYSIS OF INDIVIDUAL POSITIVE AND NEGATIVE PULSES

A. Simultaneous Current and Electric Field Waveforms Fig. 4 presents the plots of the individual waveforms of current and electric field associated with the positive pulse of the bipolar flash. Fig. 5 and 6 present similar plots for two negative return strokes of the same flash.

As discussed in [START_REF] Azadifar | Analysis of lightning-ionosphere interaction using simultaneous records of source current and 380 km distant electric field[END_REF], a 60-kHz ringing can be seen in all the measured E-field waveforms. This effect might be due to the cable-car grounding wires of about 2.3 km in length connecting the Säntis Tower to the mountain base, which could form a halfwave resonator at that frequency [START_REF] Azadifar | Analysis of lightning-ionosphere interaction using simultaneous records of source current and 380 km distant electric field[END_REF].

B. Ionospheric Reflections

Using the zero-to-zero approach [START_REF] Haddad | New measurements of lightning electric fields in Florida: Waveform characteristics, interaction with the ionosphere, and peak current estimates[END_REF], one can infer the height of the reflection for the positive pulse to be about 94 km. It should be noted that this flash occurred during daytime when the average height of reflection should be about 77 km. This flash contains numerous other negative pulses, both before and after the positive one. Fig. 7 shows, sequentially, the evaluated reflection heights for pulses of this flash that exhibited ionospheric reflections. As it can be seen, the reflection height of the positive pulse (94.9 km) is significantly higher than those for negative ICC pulses and return strokes in this flash, which range from 73 to 81 km.

Changes of ionospheric reflection height for first and subsequent return strokes in downward flashes were reported in [START_REF] Haddad | New measurements of lightning electric fields in Florida: Waveform characteristics, interaction with the ionosphere, and peak current estimates[END_REF], [START_REF] Somu | A study of changes in apparent ionospheric reflection height within individual lightning flashes[END_REF]. They also observed that the reflection height tends to increase with increasing return-stroke peak current. Azadifar et al. [START_REF] Azadifar | Analysis of lightning-ionosphere interaction using simultaneous records of source current and 380 km distant electric field[END_REF], in their analysis of negative upward flashes initiated from the Säntis Tower, did not observe any significant variation of the reflection height as a function of stroke order within a flash. As discussed in [START_REF] Azadifar | Analysis of lightning-ionosphere interaction using simultaneous records of source current and 380 km distant electric field[END_REF], this might be due to the fact that the peak currents in their recorded upward flashes were much smaller than the downward flashes studied by Somu et al. [START_REF] Somu | A study of changes in apparent ionospheric reflection height within individual lightning flashes[END_REF].

In our dataset (see Fig. 7), the reflection height inferred for negative strokes before and after the positive pulse assumes very similar values, while the estimated reflection height for the positive pulse is significantly higher. This is to some extent consistent with the analysis of [START_REF] Haddad | New measurements of lightning electric fields in Florida: Waveform characteristics, interaction with the ionosphere, and peak current estimates[END_REF], [START_REF] Somu | A study of changes in apparent ionospheric reflection height within individual lightning flashes[END_REF], given that the return stroke current peaks for negative strokes range from 2.9 to 13.2 kA, while the positive pulse peak is about 40 kA. More data and further analyses are needed to understand the dependence of the reflection height on the source characteristics (return stroke current peak, frequency content, polarity, channel geometry, etc.).

C. Field-to-Current Conversion Factor

Fig. 8 shows a scatterplot of electric field peak versus current peak for all the 24 detected pulses. Negative pulses are shown in blue and the measured positive one is shown in red in this figure. The best fit linear regression (forced to go through the origin of coordinates) is also shown in Fig. 8.

It can be seen that the ratio of the peak field to the current peak is much smaller (factor of about 2.3) for the positive pulse compared to negative pulses. Among reasons that could explain this difference are:

-The return stroke speed for positive strokes is, in general, smaller than that for negative strokes [START_REF] Wang | Observed leader and return-stroke propagation characteristics in the bottom 400 m of a rocket-triggered lightning channel[END_REF]- [START_REF] Idone | The propagation speed of a positive lightning return stroke[END_REF]; -The enhancement of the electric field due to the presence of the tower and the mountain [START_REF] Azadifar | Evaluation of the performance characteristics of the European Lightning Detection Network EUCLID in the Alps region for upward negative flashes using direct measurements at the instrumented Säntis Tower[END_REF]- [START_REF] Li | On Lightning Electromagnetic Field Propagation Along an Irregular Terrain[END_REF] might be more significant for negative pulses, which are characterized by faster risetimes, than for positive pulses.

A recent full wave theoretical analysis of the propagation effects of lightning electromagnetic fields over mountainous terrain [START_REF] Li | On Lightning Electromagnetic Field Propagation Along an Irregular Terrain[END_REF], supported by experimental observations consisting of simultaneous records of lightning currents and electric fields associated with upward negative lightning flashes to the instrumented Säntis Tower revealed that the combined effect of the tower and the mountainous terrain topography around the Säntis Tower results in an enhancement of the radiated electric field from negative return strokes. Further full-wave analysis is needed to investigate the effect of these factors on the propagation of radiated electromagnetic fields associated with positive return strokes. V. CONCLUSION

We presented and discussed simultaneous records of current and wideband electric field waveforms at 380 km distance from the strike point associated with a bipolar upward flash initiated from the Säntis Tower. The recorded flash contains 23 negative and one positive strokes.

The height of the ionospheric reflection for the positive pulse was inferred to be about 94.9 km, a value which is significantly higher than for negative pulses of this same flash, which range from 73.3 to 80.9 km.

It is also found that the ratio of the peak field to the current peak is about two times smaller for the positive pulse compared to negative pulses. This difference can be attributed to a lower return stroke speed for the positive stroke, compared to the negative strokes, and also to the fact that the enhancement of the electric field due to the presence of the tower and the mountain might be more significant for negative pulses, which are characterized by faster risetimes, than for the positive pulse.

Fig. 1 .

 1 Fig. 1. Monthly distribution of negative, positive and bipolar lightning flashes recorded at Säntis Tower over the period ranging from June 2010 to July 2016.

Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .

 23456 Fig. 2. The locationa of the Säntis Tower (Switzerland) and the electric field measurement station at Neudorf, Austria. The distance between Neudorf and the Säntis is 380 km.

Fig. 7 .

 7 Fig. 7. Sequence of evaluated reflection heights for the negative pulses (in red) and the positive pulse (in blue) of the flash of 21 September 2014, 15:14:09.

Fig. 8 .

 8 Fig. 8. Electric field peak of the ground wave at 380 km versus current peak for all the pulses of the flash of 21 September 2014, 15:14:09. The positive pulse is shown in red and 23 negative pulses are shown in blue. The best fit linear regression is also shown in the figure.

ACKNOWLEDGMENT Financial supports from the Swiss National Science Foundation (Project No. 200021_147058) and the European Union's Horizon 2020 research and innovation programme (grant agreement. No 737033-LLR) are acknowledged.