Nohra Hage

Coherent presentations of super plactic monoids of type A by insertions

Keywords: Super plactic monoids, syzygies, super tableaux, insertions, string rewriting. M.S.C 20M05, 16S15. Secondary: 68Q42, 20M35

In this paper, we study by rewriting methods the presentations of the super plactic monoid of type A which is related to the representations of the general linear Lie superalgebra. We construct a convergent presentation of this monoid whose generators are super columns and whose rules are defined by insertions on super tableaux over a signed alphabet. We extend this presentation into a coherent one whose syzygies are defined as relations among insertion algorithms. Finally, we reduce this coherent presentation to a Tietze equivalent one over the initial signed alphabet. Such coherent presentations are used for representations of super plactic monoids by describing their actions on categories.

Introduction

We study the presentations of the super version of the plactic monoid of type A over a signed alphabet by rewriting methods using combinatorial properties of super tableaux. Study presentations by rewriting consists in the orientation of the relations, then called reduction rules. Two words are equal in a monoid presented by a rewriting system if they are related by a zig-zag sequence of applications of reductions rules. A rewriting system is convergent if the reduction relation induced by the rules is well-founded and any reductions starting on a same word can be extended to end on a same reduced word. Starting with a presentation of a monoid, we are interested in the computation of all syzygies for this presentation and in particular to compute a family of generators for the syzygies. In commutative algebra, the theory of Gröbner bases gives algorithms to compute bases for linear syzygies. By a similar method, the syzygy problem for presentation of monoids can be algorithmically solved using coherent convergent presentations. Such presentations extend the notion of a convergent presentation of the monoid by globular homotopy generators taking into account the relations amongst the relations. Study the syzygies in a monoid produces in higher dimensions free objects that are homotopically equivalent to the original monoid and then allows us to compute its homological invariants. Indeed, this study provides the first two steps in the computation of a polygraphic resolution of the monoid, that is, a categorical cofibrant replacement of the monoid in a free (𝜔, 1)-category, whose acyclicity is proved by an iterative construction of a normalization reduction strategy, [START_REF] Guiraud | Higher-dimensional normalisation strategies for acyclicity[END_REF]. Moreover, coherent presentations are used to describe the notion of actions of the monoid on categories, [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF].

This paper is a part of a broader project that consists of studying, by a rewriting approach, families of plactic-like monoids defined from combinatorial objects constructed using insertion algorithms. For instance, plactic monoids are related to Young tableaux, [START_REF] Lascoux | ), volume 109 of Quad[END_REF][START_REF] Lecouvey | Combinatorics of crystal graphs for the root systems of 𝐴 𝑛 , 𝐵 𝑛 , 𝐶 𝑛 , 𝐷 𝑛 and 𝐺 2[END_REF], Chinese monoids to Chinese staircases, [START_REF] Cassaigne | The Chinese monoid[END_REF], hypoplactic monoids to quasi-ribbon tableaux, [START_REF] Novelli | On the hypoplactic monoid[END_REF], patience sorting monoids to patience sorting tableaux, [START_REF] Cain | The monoids of the patience sorting algorithm[END_REF], and stalactic monoids to stalactic tableaux [START_REF] Priez | Lattice of combinatorial hopf algebras: binary trees with multiplicities[END_REF]. Moreover, binary search trees are used to describe normal forms for sylvester, taiga and Baxter monoids, [START_REF] Giraudo | Algebraic and combinatorial structures on pairs of twin binary trees[END_REF][START_REF] Hivert | The algebra of binary search trees[END_REF][START_REF] Priez | Lattice of combinatorial hopf algebras: binary trees with multiplicities[END_REF]. Coherent presentations are constructed for Artin monoids in [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF], for plactic monoids of type A in [START_REF] Hage | Knuth's coherent presentations of plactic monoids of type A[END_REF], for plactic monoids of type C in [START_REF] Meha | C-trees and a coherent presentation for the plactic monoid of type C[END_REF], and for Chinese monoids in [START_REF] Hage | Chinese syzygies by insertions[END_REF].

A signed alphabet is a finite or countable totally ordered set S which is disjoint union of two subsets S 0 and S 1 . The super plactic monoid over a signed alphabet S, denoted by P(S), is presented by the rewriting system Knuth 2 (S) on S submitted to the following family of super Knuth-like relations, [START_REF] Roberto | Super RSK-algorithms and super plactic monoid[END_REF]:

1. Introduction Lemma 2.6 that for all 𝑢 and 𝑣 in col(S) such that the topmost juxtaposition of ⟦𝑢⟧ 𝑟 and ⟦𝑣⟧ 𝑟 does not form a super tableau, the super tableau ⟦𝑢𝑣⟧ 𝑟 contains at most two columns, and if it contains exactly two columns then the left one contains more elements than 𝑢.

We construct in Section 2 a convergent presentation of the monoid P(S), denoted by Col 2 (S), whose set of generators is Col 1 (S) := 𝑐 𝑢 𝑢 ∈ col(S) and whose rules are 𝛾 𝑢,𝑣 : 𝑐 𝑢 𝑐 𝑣 ⇒ 𝑐 𝑤 𝑐 𝑤 ′ for every 𝑐 𝑢 , 𝑐 𝑣 in Col 1 (S) such that the topmost juxtaposition of ⟦𝑢⟧ 𝑟 and ⟦𝑣⟧ 𝑟 does not form a super tableau and where 𝑤 and 𝑤 ′ denote the readings of the left and right columns of ⟦𝑢𝑣⟧ 𝑟 respectively. We show that Col 2 (S) can be obtained from Knuth 2 (S) by applying Tietze transformations that consist in adding or removing derivable generators and in adding or removing derivable relations on a presentation of a monoid in such a way that they do not change the presented monoid. We also show that the confluence of Col 2 (S) is a direct consequence of the associativity of the product ★ 𝑟 . Theorem 2.12 states that Col 2 (S) is a convergent presentation of the monoid P(S), called the super column presentation.

We extend in Section 3 the super column presentation into coherent presentations of the monoid P(S) using the homotopical completion-reduction procedure, [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF]. Denote by Col 3 (S) the extended presentation of P(S) obtained from Col 2 (S) by adjunction of one family of syzygies of the following form is a coherent presentation of the monoid P(S), called the super column coherent presentation. We then apply the homotopical reduction procedure on Col 3 (S) in order to reduce it into a smaller one. As a first step, we apply a homotopical reduction on Col 3 (S) with a collapsible part defined by some of the generating triple confluences of Col 2 (S) and we reduce it into the coherent presentation Col 3 (S) of P(S), whose underlying rewriting system is Col 2 (S) and the syzygies X 𝑥,𝑣,𝑡 are those of Col 3 (S), but with 𝑥 in S. In a second step, we reduce Col 3 (S) into a coherent presentation PreCol 3 (S) of P(S), using a collapsible part defined by a set of syzygies of Col 3 (S). In a final step, we reduce PreCol 3 (S) into an extended presentation of P(S), denoted by Knuth 3 (S), whose underlying rewriting system is Knuth 2 (S). Theorem 3.12 states that Knuth 3 (S) is a coherent presentation of the monoid P(S).

𝑐 𝑒 𝑐 𝑒 ′𝑐 𝑡 𝑐 𝑒 𝛾 𝑒 ′ ,𝑡
Finally, we use in Subsection 3.13 this coherent presentation in order to describe the actions of super plactic monoids on categories.

Notation. Let A be a totally ordered alphabet. We will denote by A * the free monoid of words over A, the product being concatenation of words, and the identity being the empty word. We will denote by 𝑤 = 𝑥

Super column presentation of the super plactic monoid

In this article, rewriting methods are presented in the language of polygraphs, that we recall in this section and we refer the reader to [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF][START_REF] Guiraud | Polygraphs of finite derivation type[END_REF] for a deeper presentation. We also recall the notions of super plactic monoids and super tableaux from [START_REF] Hage | Super jeu de taquin and combinatorics of super tableaux of type A[END_REF][START_REF] Roberto | Super RSK-algorithms and super plactic monoid[END_REF]. We end this section by constructing a convergent presentation of the super plactic monoid by adding super columns generators and using combinatorial properties of super tableaux.

2.1. Presentations of monoids by 2-polygraphs. In this article, we deal with presentations of monoids by rewriting systems, described by 2-polygraphs with only 0-cell denoted by •. Such a 2polygraph Σ is given by a pair (Σ 1 , Σ 2 ), where Σ 1 is a set and Σ 2 is a globular extension of the free monoid

Σ * 1 , that is a set of 2-cells 𝛼 : 𝑠 1 (𝛼) ⇒ 𝑡 1 (𝛼) relating 1-cells in Σ * 1 ,
where 𝑠 1 (𝛼) and 𝑡 1 (𝛼) denote the source and the target of 𝛼 respectively. If there is no possible confusion, Σ 2 will denote the 2-polygraph itself. Recall that a 2-category (resp. (2, 1)-category) is a category enriched in categories (resp. in groupoids). When two 1-cells, or 2-cells, 𝑓 and 𝑔 of a 2-category are 0-composable (resp. 1-composable), we denote by 𝑓 𝑔 (resp. 𝑓 ★ 1 𝑔) their 0-composite (resp. 1-composite). We will denote by Σ * 2 (resp. Σ ⊤ 2 ) the 2-category (resp. (2, 1)-category) freely generated by the 2-polygraph Σ, see [START_REF] Guiraud | Polygraphs of finite derivation type[END_REF] for more information. The monoid presented by a 2-polygraph Σ, denoted by Σ, is defined as the quotient of the free monoid Σ * 1 by the congruence generated by the set of 2-cells Σ 2 . A presentation of a monoid M is a 2-polygraph whose presented monoid is isomorphic to M. Two 2-polygraphs are Tietze equivalent if they present isomorphic monoids.

A 2-cell 𝛼 of a 2-polygraph Σ is collapsible, if 𝑡 1 (𝛼) is a 1-cell of Σ 1 and 𝑠 1 (𝛼) does not contain 𝑡 1 (𝛼), then 𝑡 1 (𝛼) is called redundant. An elementary Tietze transformation of a 2-polygraph Σ is a 2-functor with domain Σ ⊤
2 that belongs to one of the following transformations, [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF]:

i) adjunction 𝜄 1 𝛼 : Σ ⊤ 2 → Σ ⊤ 2 [𝑥] (𝛼) (resp. 𝜄 𝛼 : Σ ⊤ 2 → Σ ⊤ 2 (𝛼)) of a redundant 1-cell 𝑥 with its collapsible 2-cell 𝛼 (resp. of a redundant 2-cell 𝛼), ii) elimination 𝜋 𝛼 : Σ ⊤ 2 → (Σ 1 \ {𝑥 }, Σ 2 \ {𝛼 }) ⊤ (resp. 𝜋 (𝜇,𝛼) : Σ ⊤ 2 → Σ ⊤ 2 /(𝜇, 𝛼)) of a redundant 1-cell 𝑥 with its collapsible 2-cell 𝛼 (resp. of a redundant 2-cell 𝛼).
If Σ and Υ are 2-polygraphs, a Tietze transformation from Σ to Υ is a 2-functor 𝐹 : Σ ⊤ → Υ ⊤ that decomposes into sequence of elementary Tietze transformations. Two 2-polygraphs are Tietze equivalent if and only if there exists a Tietze transformation between them, [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF]. Given a 2-polygraph Σ and a

2-cell 𝜇 1 ★ 1 𝜇 ★ 1 𝜇 2 in Σ ⊤ 2 ,
(𝑓 ′ ) = 𝑡 1 (𝑓 ), 𝑠 1 (𝑔 ′ ) = 𝑡 1 (𝑔) and 𝑡 1 (𝑓 ′ ) = 𝑡 1 (𝑔 ′ ). A 2-polygraph Σ is confluent if all of
its branchings are confluent. It is convergent if it terminates and it is confluent. In that case, every 1-cell 𝑢 of Σ * 1 has a unique normal form.

Super Young tableaux and insertions.

A partition of a positive integer 𝑛, is a weakly decreasing sequence 𝜆 = (𝜆 1 , . . . , 𝜆 𝑘 ) ∈ N 𝑘 such that 𝜆 𝑖 = 𝑛. The integer 𝑘 is called number of parts or height of 𝜆. The Young diagram of a partition 𝜆 = (𝜆 1 , . . . , 𝜆 𝑘 ) is the set Y (𝜆) := (𝑖, 𝑗) 1 ⩽ 𝑖 ⩽ 𝑘, 1 ⩽ 𝑗 ⩽ 𝜆 𝑖 , that can be represented by a diagram by drawing a box for each pair (𝑖, 𝑗). The transposed diagram {( 𝑗, 𝑖) (𝑖, 𝑗) ∈ Y (𝜆)} defines another partition, called the conjugate partition of 𝜆, whose parts are the lengths of the columns of Y (𝜆). Let 𝜆 be a partition. A super semistandard Young tableau, or super tableau for short, over S is a pair 𝑡 := (𝜆, T ) where T : Y (𝜆) → S is a map satisfying T (𝑖, 𝑗) ⩽ T (𝑖, 𝑗 + 1), with T (𝑖, 𝑗) = T (𝑖, 𝑗 + 1) only if || T (𝑖, 𝑗)|| = 0, and T (𝑖, 𝑗) ⩽ T (𝑖 + 1, 𝑗), with T (𝑖, 𝑗) = T (𝑖 + 1, 𝑗) only if || T (𝑖, 𝑗)|| = 1. We will call Y (𝜆), T and 𝜆, the frame, the filing and the shape of the super tableau 𝑡 respectively. We will denote by Yt(S) the set of all super tableaux over S.

Denote by 𝑅 𝑟𝑜𝑤 (resp. 𝑅 𝑐𝑜𝑙 ) the reading map on Yt(S) that reads a super tableau row-wise (columnwise) from bottom to top and from left to right. For instance, consider the alphabet S = {1, 2, 3, 4, 5} with signature given by S 0 = {1, 2, 4} and S 1 defined consequently. The following diagram is a super tableau over S:

𝑡 = 1 1 2 3 4 4 5 5
with 𝑅 𝑟𝑜𝑤 (𝑡) = 55344112 and 𝑅 𝑐𝑜𝑙 (𝑡) = 55314142.

A super row (resp. super column) is a word 𝑥 1 . . . 𝑥 𝑘 in S * such that 𝑥 𝑖 ⩽ 𝑥 𝑖+1 (resp. 𝑥 𝑖+1 ⩽ 𝑥 𝑖 ) with 𝑥 𝑖 = 𝑥 𝑖+1 only if ||𝑥 𝑖 || = 0 (resp. ||𝑥 𝑖 || = 1). In other words, a super row (resp. super column) is the reading of a super tableau whose shape is a row (resp. column). We will denote by col(S) the set of all super columns over S. Denote by ≼ deglex (resp. ≼ rev ) the length lexicographic order (resp. length reverse lexicographic order) on col(S) defined by 𝑢 ≼ deglex 𝑣 (resp.

𝑢 ≼ rev 𝑣) if |𝑢 | < |𝑣 | (resp. |𝑢 | > |𝑣 |) or |𝑢 | = |𝑣 |
and 𝑢 < 𝑙𝑒𝑥 𝑣, for all 𝑢 and 𝑣 in col(S), where < lex denotes the lexicographic order on S.

Recall from [START_REF] Roberto | Super RSK-algorithms and super plactic monoid[END_REF] the right and left insertion algorithms on Yt(S) that insert an element 𝑥 in S into a super tableau 𝑡 of Yt(S). The right (or row) insertion, denoted by ⇝ , computes a super tableau 𝑡 ⇝ 𝑥 as follows. If 𝑥 ∈ S 𝑜 (resp. 𝑥 ∈ S 1 ) is at least as large as (resp. larger than) the last element of the top row of 𝑡, then put 𝑥 in a box to the right of this row. Otherwise, let 𝑦 be the smallest element of the top row of 𝑡 such that 𝑦 > 𝑥 (resp. 𝑦 ⩾ 𝑥). Then 𝑥 replaces 𝑦 in this row and 𝑦 is bumped into the next row where the process is repeated. The algorithm terminates when the element which is bumped is at least as large as (resp. larger than) the last element of the next row. Then it is placed in a box at the right of that row. The left (or column) insertion, denoted by ⇝, computes a super tableau 𝑥 ⇝ 𝑡 as follows.

If 𝑥 ∈ S 𝑜 (resp. 𝑥 ∈ S 1 ) is larger than (resp. at least as large as) the bottom element of the leftmost column of 𝑡, then put 𝑥 in a box to the bottom of this column. Otherwise, let 𝑦 be the smallest element of the leftmost column of 𝑡 such that 𝑦 ⩾ 𝑥 (resp. 𝑦 > 𝑥). Then 𝑥 replaces 𝑦 in this column and 𝑦 is bumped into the next column where the process is repeated. The algorithm terminates when the element which is bumped is greater than (resp. at least as large as) all the elements of the next column. Then it is placed in a box at the bottom of that column. For instance, consider S = N with signature given by S 0 the set of even numbers and S 1 defined consequently. We have

1 2 2 3 1 3 4 3 ⇝ 2 = 1 2 2 2
⟦𝑤⟧ 𝑟 := (∅ ⇝ 𝑤) = ((. . . (∅ ⇝ 𝑥 1 ) ⇝ . . .) ⇝ 𝑥 𝑘 ).
Note that for any super tableau 𝑡 in Yt(S), the equality ⟦𝑅 𝑟𝑜𝑤 (𝑡)⟧ 𝑟 = 𝑡 holds in Yt(S), [START_REF] Roberto | Super RSK-algorithms and super plactic monoid[END_REF]. We define an internal product ★ 𝑟 on Yt(S) by setting

𝑡 ★ 𝑟 𝑡 ′ := (𝑡 ⇝ 𝑅 𝑟𝑜𝑤 (𝑡 ′ ))
for all 𝑡, 𝑡 ′ in Yt(S). By definition the relations 𝑡 ★ 𝑟 ∅ = 𝑡 and ∅ ★ 𝑟 𝑡 = 𝑡 hold, showing that the product ★ 𝑟 is unitary with respect to ∅.

2.3.

The super plactic monoid. The super plactic monoid over S, denoted by P(S), is presented by the 2-polygraph Knuth 2 (S), whose set of 1-cells is S and whose 2-cells are, [START_REF] Roberto | Super RSK-algorithms and super plactic monoid[END_REF] :

𝜂 𝑥,𝑦,𝑧 : 𝑧𝑥𝑦 ⇒ 𝑥𝑧𝑦, with 𝑥 = 𝑦 only if ||𝑦|| = 0 and 𝑦 = 𝑧 only if ||𝑦|| = 1, 𝜀 𝑥,𝑦,𝑧 : 𝑦𝑧𝑥 ⇒ 𝑦𝑥𝑧, with 𝑥 = 𝑦 only if ||𝑦|| = 1 and 𝑦 = 𝑧 only if ||𝑦|| = 0, (1)
oriented with respect the lexicographic order, for any 𝑥 ⩽ 𝑦 ⩽ 𝑧 of elements of S. The congruence generated by this 2-polygraph, denoted by ∼ P( S) , is called the super plactic congruence. Note that since the relations (1) are Z 2 -homogeneous we have that P(S) is a supermonoid. Moreover, for any 𝑤 in S * , we have 𝑤 ∼ P( S) 𝑅 𝑟𝑜𝑤 (⟦𝑤⟧ 𝑟 ), [START_REF] Roberto | Super RSK-algorithms and super plactic monoid[END_REF], and for any 𝑡 in Yt(S), we have 𝑅 𝑟𝑜𝑤 (𝑡) ∼ P( S) 𝑅 𝑐𝑜𝑙 (𝑡), [START_REF] Hage | Super jeu de taquin and combinatorics of super tableaux of type A[END_REF]. Note finally that super tableaux satisfy the cross-section property for ∼ P( S) , that is, for all 𝑤 and 𝑤 ′ in S * , 𝑤 ∼ P( S) 𝑤 ′ if and only if ⟦𝑤⟧ 𝑟 = ⟦𝑤 ′ ⟧ 𝑟 , [START_REF] Roberto | Super RSK-algorithms and super plactic monoid[END_REF]. As a consequence of the cross-section property, we deduce that the product ★ 𝑟 is associative and the following equality

𝑦 ⇝ (𝑡 ⇝ 𝑥) = (𝑦 ⇝ 𝑡) ⇝ 𝑥 (2)
2.5. Graphical notations holds in Yt(S), for all 𝑡 in Yt(S) and 𝑥, 𝑦 in S. In particular, for any word 𝑤 = 𝑥 1 . . . 𝑥 𝑘 in S * the super tableau ⟦𝑤⟧ 𝑟 can be also computed by inserting its elements iteratively from right to left using the left insertion starting from the empty tableau:

⟦𝑤⟧ 𝑟 = (𝑤 ⇝ ∅) := (𝑥 1 ⇝ (. . . ⇝ (𝑥 𝑘 ⇝ ∅) . . .)).
Note that the associativity of the product ★ 𝑟 can be obtained using the properties of the super jeu de taquin introduced in [START_REF] Hage | Super jeu de taquin and combinatorics of super tableaux of type A[END_REF]. Note also that we can show the equality (2) using only the definitions of the insertion algorithms and without supposing the cross-section property, and then the associativity of the product ★ 𝑟 and the cross-section property will be consequences of this equality.

Let 𝑤 be in S * . For any 𝑘 ⩾ 0, denote by 𝑙 𝑘 (𝑤) (resp. 𝑙 𝑘 (𝑤)) the maximal number which can be obtained as the sum of the lengths of 𝑘 super rows (resp. super columns) that are disjoint subsequences of 𝑤. Let 𝜆 = (𝜆 1 , . . . , 𝜆 𝑘 ) be the shape of the super tableau ⟦𝑤⟧ 𝑟 and 𝜆 = ( 𝜆 1 , . . . , 𝜆 𝑙 ) be the conjugate partition of 𝜆. For any 𝑘 ⩾ 0, we have 𝑙 𝑘 (𝑤) = 𝜆 1 + . . . + 𝜆 𝑘 and 𝑙 𝑘 (𝑤) = 𝜆 1 + . . . + 𝜆 𝑙 , [START_REF] Roberto | Super RSK-algorithms and super plactic monoid[END_REF]. In particular, we deduce the following result.

Lemma ([24]

). Let 𝑤 be a word over S. The number of columns in the super tableau ⟦𝑤⟧ 𝑟 is equal to the length of the longest super row word that is a subsequence of 𝑤, and the number of rows in ⟦𝑤⟧ 𝑟 is equal to the length of the longest super column word that is a subsequence of 𝑤.

Graphical notations.

Given 𝑢 = 𝑥 1 . . . 𝑥 𝑝 and 𝑣 = 𝑦 1 . . . 𝑦 𝑞 in col(S). We will use the following notations depending on whether the juxtaposition of ⟦𝑢⟧ 𝑟 and ⟦𝑣⟧ 𝑟 forms a super tableau: , from the bigger column to the smaller one with respect to ≼ deglex . The composite

𝑇 𝜇←𝛾 ′ = 𝜅 ′ 𝜇 𝑥 3 𝑥 2 𝑥 1 ←𝛾 ′ 𝑥 3 ,𝑥 2 𝑥 1 • . . . • 𝜅 ′ 𝜇 𝑥𝑝 ...𝑥 1 ←𝛾 ′ 𝑥𝑝 ,𝑥 𝑝-1 ...𝑥

Coherent presentations of the super plactic monoid

In this section, we begin by recalling the notion of coherent presentations of monoids from [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF] and we extend the super column presentation into a coherent presentation of the super plactic monoid. In a second part, we recall the homotopical reduction procedure from [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF] and we reduce the super coherent column presentation into a smaller one over the initial signed alphabet. We follow for this aim the approach developed in [START_REF] Hage | Knuth's coherent presentations of plactic monoids of type A[END_REF] for the non-signed case.

(𝐴) = 𝑠 1 𝑡 2 (𝐴) and 𝑡 1 𝑠 2 (𝐴) = 𝑡 1 𝑡 2 (𝐴)
. Such a 3-cell can be represented with the following globular shape:

• 𝑢 " " 𝑣 < < 𝑓 Õ % 𝑔 Õ % 𝐴 7 W • or 𝑢 𝑓 ! A 𝑔 R r 𝑣 𝐴 Õ %
We will denote by Σ ⊤ 3 the free (3, 1)-category generated by the

(3, 1)-polygraph (Σ 2 , Σ 3 ). A pair (𝑓 , 𝑔) of 2-cells of Σ ⊤ 2 such that 𝑠 1 (𝑓 ) = 𝑠 1 (𝑔) and 𝑡 1 (𝑓 ) = 𝑡 1 (𝑔) is called a 2-sphere of Σ ⊤ 2 .
An extended presentation of a monoid M is a (3, 1)-polygraph whose underlying 2-polygraph is a presentation of the monoid M. A coherent presentation of M is an extended presentation Σ of M such that the cellular extension Σ 3 is a homotopy basis of the (2, 1)-category Σ ⊤ 2 , that is, for every 2-sphere 𝜇 of Σ ⊤ 2 , there exists a 3-cell in Σ ⊤ 3 with boundary 𝜇. The elements in Σ ⊤ 3 are called syzygies of the presentation Σ. Let Σ be a (3, 1)-polygraph. A 3-cell 𝐴 of Σ is called collapsible if 𝑡 2 (𝐴) is in Σ 2 and 𝑠 2 (𝐴) is a 2-cell of the free (2, 1)-category over (Σ 2 \ {𝑡 2 (𝐴)}) ⊤ , then 𝑡 2 (𝐴) is called redundant. An elementary Tietze transformation of a (3, 1)-polygraph Σ is a 3-functor with domain Σ ⊤ 3 that belongs to one of the following operations, [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF]:

i) adjunction 𝜄 1
𝛼 and elimination 𝜋 𝛼 of a 2-cell 𝛼 as described in Subsection 2.1,

ii) coherent adjunction 𝜄 2 𝐴 : Σ ⊤ 3 → Σ ⊤ 3 (𝛼) (𝐴) (resp. 𝜄 𝐴 : Σ ⊤ 3 → Σ ⊤ 3 (𝐴)) of a redundant 2-cell 𝛼 with its collapsible 3-cell 𝐴 (resp. of a redundant 3-cell 𝐴), iii) coherent elimination 𝜋 𝐴 : Σ ⊤ 3 → Σ ⊤ 3 /𝐴 (resp. 𝜋 (𝐵,𝐴) : Σ ⊤ 3 → Σ ⊤ 3 /(𝐵, 𝐴)
) of a redundant 2-cell 𝛼 with its collapsible 3-cell 𝐴 (resp. of a redundant 3-cell 𝐴, that maps 𝐴 to 𝐵).

For (3, 1)-polygraphs Σ and Υ, a Tietze transformation from Σ to Υ is a 3-functor 𝐹 : Σ ⊤ 3 → Υ ⊤ 3 that decomposes into a sequence of elementary Tietze transformations. Two (3, 1)-polygraphs Σ and Υ are Tietze-equivalent if there exists an equivalence of 2-categories 𝐹 : Σ ⊤ 2 /Σ 3 → Υ ⊤ 2 /Υ 3 and the presented monoids Σ 2 and Υ 2 are isomorphic. Two (3, 1)-polygraphs are Tietze equivalent if, and only if, there exists a Tietze transformation between them, [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF].

Recall that Squier's coherence theorem, [START_REF] Squier | A finiteness condition for rewriting systems[END_REF], states that, any convergent 2-polygraph Σ presenting a

Coherent presentations of the super plactic monoid

monoid M can be extended into a coherent presentation of M having a generating syzygy

𝑣 𝑓 ′ # C 𝐴 Õ % 𝑢 𝑓 6 V 𝑔 7 W 𝑤 𝑣 ′ 𝑔 ′ Q q
for every critical branching (𝑓 , 𝑔) of Σ, where 𝑓 ′ and 𝑔 ′ are chosen confluent rewriting paths.

3.2. Super column coherent presentation. Denote by Col 3 (S) the extended presentation of the monoid P(S) obtained from Col 2 (S) by adjunction of one family of syzygies X 𝑢,𝑣,𝑡 of the following form

𝑐 𝑒 𝑐 𝑒 ′𝑐 𝑡 𝑐 𝑒 𝛾 𝑒 ′ ,𝑡 7 W X 𝑢,𝑣,𝑡 Õ % 𝑐 𝑒 𝑐 𝑏 𝑐 𝑏 ′ 𝛾 𝑒,𝑏 𝑐 𝑏 ′ ' G 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 𝑐 𝑡 9 Y 𝑐 𝑢 𝛾 𝑣,𝑡 5 U 𝑐 𝑎 𝑐 𝑑 𝑐 𝑏 ′ 𝑐 𝑢 𝑐 𝑤 𝑐 𝑤 ′ 𝛾 𝑢,𝑤 𝑐 𝑤 ′ 7 W 𝑐 𝑎 𝑐 𝑎 ′𝑐 𝑤 ′ 𝑐 𝑎 𝛾 𝑎 ′ ,𝑤 ′ F f (4)
for every 𝑢, 𝑣 and 𝑡 in Col(S) such that 𝑢 𝑣 𝑡 × × , where 𝑒 and 𝑒 ′ (resp. 𝑤 and 𝑤 ′ ) are the readings of the two columns of the super tableau ⟦𝑢𝑣⟧ 𝑟 (resp. ⟦𝑣𝑡⟧ 𝑟 ), and where 𝑎 and 𝑎 ′ (resp. 𝑏 and 𝑏 ′ ) denote the readings of the two columns of the super tableau ⟦𝑢𝑤⟧ 𝑟 (resp. ⟦𝑒 ′ 𝑡⟧ 𝑟 ) and 𝑎, 𝑑, 𝑏 ′ are the readings of the three columns of the super tableau ⟦𝑢𝑣𝑡⟧ 𝑟 . Note that one or further columns 𝑒 ′ , 𝑤 ′ , 𝑎 ′ and 𝑏 ′ can be empty as illustrated in Figure 1. In those cases some indicated 2-cells 𝛾 in the confluence diagram correspond to identities.

) I C 𝑢,𝑣,𝑡 Õ % 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 𝑐 𝑡 7 W 𝑐 𝑢 𝛾 𝑣,𝑡 6 V 𝑐 𝑢𝑤 𝑐 𝑤 ′ 𝑐 𝑢 𝑐 𝑤 𝑐 𝑤 ′ 𝛾 𝑢,𝑤 𝑐 𝑤 ′ D d 𝑢 𝑣 𝑡 ×1 ×2 with 𝑢𝑣 𝑡 𝑐 𝑢𝑣 𝑐 𝑡 C ′ 𝑢,𝑣,𝑡 Õ % 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 𝑐 𝑡 7 W 𝑐 𝑢 𝛾 𝑣,𝑡 6 V 𝑐 𝑢 𝑐 𝑤 𝑐 𝑤 ′ 𝛾 𝑢,𝑤 𝑐 𝑤 ′ 7 W 𝑐 𝑢𝑣 𝑐 𝑎 ′𝑐 𝑤 ′ 𝑐 𝑢𝑣 𝛾 𝑎 ′ ,𝑤 ′ u 𝑢 𝑣 𝑡 ×2 ×2 𝑐 𝑒 𝑐 𝑒 ′𝑐 𝑡 𝑐 𝑒 𝛾 𝑒 ′ ,𝑡 7 W D 𝑢,𝑣,𝑡 Õ % 𝑐 𝑒 𝑐 𝑏 𝑐 𝑏 ′ 𝛾 𝑒,𝑏 𝑐 𝑏 ′ ( H 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 𝑐 𝑡 8 X 𝑐 𝑢 𝛾 𝑣,𝑡 5 U 𝑐 𝑎 𝑐 𝑑 𝑐 𝑏 ′ 𝑐 𝑢 𝑐 𝑤 𝑐 𝑤 ′ 𝛾 𝑢,𝑤 𝑐 𝑤 ′ 7 W 𝑐 𝑎 𝑐 𝑎 ′𝑐 𝑤 ′ 𝑐 𝑎 𝛾 𝑎 ′ ,𝑤 ′ E e
Σ ⊤ 3 is a pair (𝑓 , 𝑔) of 3-cells of Σ ⊤ 3 such that 𝑠 2 (𝑓 ) = 𝑠 2 (𝑔) and 𝑡 2 (𝑓 ) = 𝑡 2 (𝑔). A collapsible part of Σ is a triple (Γ 2 , Γ 3 , Γ 4 ) made of a family Γ 2 of 2-cells of Σ, a family Γ 3 of 3-cells of Σ and a family Γ 4 of 3-spheres of Σ ⊤
3 , such that the following conditions are satisfied, [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF]:

i) every 𝜇 of every Γ 𝑘 is collapsible, that is, 𝑡 𝑘-1 (𝜇) is in Σ 𝑘-1 and 𝑠 𝑘-1 (𝜇) does not contain 𝑡 𝑘-1 (𝜇),
ii) no cell of Γ 2 (resp. Γ 3 ) is the target of a collapsible 3-cell of Γ 3 (resp. 3-sphere of Γ 4 ),

iii) there exists a well-founded order on the cells of Σ such that, for every 𝜇 in every Γ 𝑘 , 𝑡 𝑘-1 (𝜇) is strictly greater than every generating (𝑘 -1)-cell that occurs in the source of 𝜇.

The homotopical reduction of the (3, 1)-polygraph Σ with respect to a collapsible part Γ is the Tietze transformation, denoted by 𝑅 Γ , from the (3, 1)-category Σ ⊤ 3 to the (3, 1)-category freely generated by the (3, 1)-polygraph obtained from Σ by removing the cells of Γ and all the corresponding redundant cells. For any 𝜇 in Γ, we have 𝑅 Γ (𝑡 (𝜇)) = 𝑅 Γ (𝑠 (𝜇)) and 𝑅 Γ (𝜇) = 1 𝑅 Γ (𝑠 (𝜇)) . In any other cases, the transformation 𝑅 Γ acts as an identity, see [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF] for more details.

A local triple branching of a 2-polygraph Σ is a triple (𝑓 , 𝑔, ℎ) of rewriting steps of Σ with a common source. An aspherical triple branchings have two of their 2-cells equal. A Peiffer triple branchings have at least one of their 2-cells that form a Peiffer branching with the other two. The overlap triple branchings are the remaining local triple branchings. Local triple branchings are ordered by inclusion of their sources and a minimal overlap triple branching is called critical. If Σ is a coherent and convergent (3, 1)-polygraph, a triple generating confluence of Σ is a 3-sphere

𝑣 𝑓 ′ 1 2 R 1 Q 𝑠 ′ ℎ ′′ 7 Õ % 𝑣 𝑓 ′ 1 2 R 𝑓 ′ 2 & F Õ % 𝑠 ′ ℎ ′′ 7 l Ð 𝑢 𝑓 @ `𝑔 7 W ℎ 4 T 𝑤 𝑔 ′ 1 H h 𝑔 ′ 2 & F l Ð 𝑢 Ω 𝑓 ,𝑔,ℎ 1 c 𝑢 𝑓 @ h 4 T 𝑤 ′ 𝑔 ′′ 7 W 1 Q 𝑢 𝑠 ℎ ′ 2 A a 𝑣 ′ 𝑓 ′′ X x 𝑠 ℎ ′ 1 H h ℎ ′ 2 A a 𝑣 ′ 𝑓 ′′ X x
where (𝑓 , 𝑔, ℎ) is a triple critical branching of the 2-polygraph Σ 2 and the other cells are obtained by confluence, [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF].

3.5. Reduced super coherent column presentation. We apply the homotopical reduction procedure on the (3, 1)-polygraph Col 3 (S) using the collapsible part made of generating triple confluences. Following Theorem 3.3, the family of syzygies X 𝑢,𝑣,𝑡 given in (4) and indexed by columns 𝑢, 𝑣 and 𝑡 in col(S) such that 𝑢 𝑣 𝑡 × × forms a homotopy basis of the (2, 1)-category Col 2 (S) ⊤ . Consider such a triple (𝑢, 𝑣, 𝑡) with |𝑢 | ⩾ 2, and let 𝑥 𝑝 be in S and 𝑢 1 be in col(S) such that 𝑥 𝑝 = ℓ (𝑢) and 𝑢 1 = Rem(𝑢). We will show that the confluence diagram induced by the critical triple branching with source 𝑐 𝑥 𝑝 𝑐 𝑢 1 𝑐 𝑣 𝑐 𝑡 is represented by the 3-sphere Ω 𝑥 𝑝 ,𝑢 1 ,𝑣,𝑡 whose source is the following 3-cell

𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 5 U X 𝑥 𝑝 ,𝑢 1 ,𝑣 𝑐 𝑡 Õ % 𝑐 𝑒 𝑐 𝑒 ′𝑐 𝑡 𝛾 𝑒 ′ ,𝑡 1 Q 𝑐 𝑒 X 𝑦,𝑠 ′ ,𝑡 Õ % 𝑐 𝑒 𝑐 𝑦 𝑐 𝑠 ′𝑐 𝑡 𝛾 𝑦,𝑠 ′ C c 𝛾 𝑠 ′ ,𝑡 0 P 𝑐 𝑒 𝑐 𝑏 𝑐 𝑏 ′ 𝛾 𝑒,𝑏 @ 𝑐 𝑥 𝑝 𝑐 𝑢 1 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢 1 ,𝑣 7 W 𝛾 𝑥 𝑝 ,𝑢 1 A a 𝛾 𝑣,𝑡 1 Q 𝑐 𝑥 𝑝 𝑐 𝑠 𝑐 𝑠 ′𝑐 𝑡 𝛾 𝑥 𝑝 ,𝑠 C c 𝛾 𝑠 ′ ,𝑡 0 P 𝑐 𝑥 𝑝 X 𝑢 1 ,𝑣,𝑡 Õ % ≡ 𝑐 𝑒 𝑐 𝑦 𝑐 𝑑 1 𝑐 𝑑 ′ 1 𝛾 𝑦,𝑑 1 7 W X 𝑥 𝑝 ,𝑠,𝑑 1 𝑐 𝑑 ′ 1 Õ % 𝑐 𝑒 𝑐 𝑏 𝑐 𝑠 2 𝑐 𝑑 ′ 1 𝛾 𝑠 2 ,𝑑 ′ 1 D d 𝛾 𝑒,𝑏 0 P ≡ 𝑐 𝑎 𝑐 𝑑 𝑐 𝑏 ′ 𝑐 𝑥 𝑝 𝑐 𝑢 1 𝑐 𝑤 𝑐 𝑤 ′ 𝛾 𝑢 1 ,𝑤 2 R 𝑐 𝑥 𝑝 𝑐 𝑠 𝑐 𝑑 1 𝑐 𝑑 ′ 1 𝛾 𝑠,𝑑 1 0 P 𝛾 𝑥 𝑝 ,𝑠 D d 𝑐 𝑎 𝑐 𝑑 𝑐 𝑠 2 𝑐 𝑑 ′ 1 𝛾 𝛾 𝑠 2 ,𝑑 ′ 1 S s 𝑐 𝑥 𝑝 𝑐 𝑎 1 𝑐 𝑎 ′ 1 𝑐 𝑤 ′ 𝛾 𝑎 ′ 1 ,𝑤 ′ 7 W 𝛾 𝑥 𝑝 ,𝑎 1 3 S 𝑐 𝑥 𝑝 𝑐 𝑎 1 𝑐 𝑠 3 𝑐 𝑑 ′ 1 𝛾 𝑥 𝑝 ,𝑎 1 7 W 𝑐 𝑎 𝑐 𝑧 𝑐 𝑠 3 𝑐 𝑑 ′ 1 𝛾 𝑧,𝑠 3 I i ≡ 𝑐 𝑎 𝑐 𝑧 𝑐 𝑎 ′ 1 𝑐 𝑤 ′ 𝛾 𝑎 ′ 1 ,𝑤 ′ S s

Super pre-column coherent presentation

and whose target is the following 3-cell

𝑐 𝑒 𝑐 𝑒 ′𝑐 𝑡 𝛾 𝑒 ′ ,𝑡 4 T X 𝑢,𝑣,𝑡 Õ % 𝑐 𝑒 𝑐 𝑏 𝑐 𝑏 ′ 𝛾 𝑒,𝑏 # C 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 9 Y 𝛾 𝑣,𝑡 1 Q 𝑐 𝑎 𝑐 𝑑 𝑐 𝑏 ′ 𝑐 𝑥 𝑝 𝑐 𝑢 1 𝑐 𝑣 𝑐 𝑡 𝛾 𝑥 𝑝 ,𝑢 1 @ γ𝑣,𝑡 1 Q ≡ 𝑐 𝑢 𝑐 𝑤 𝑐 𝑤 ′ 𝛾 𝑢,𝑤 7 W X 𝑥 𝑝 ,𝑢 1 ,𝑤 𝑐 𝑤 ′ Õ % 𝑐 𝑎 𝑐 𝑎 ′𝑐 𝑤 ′ 𝛾 𝑎 ′ ,𝑤 ′ D d 𝑐 𝑎 X 𝑧,𝑎 ′ 1 ,𝑤 ′ 9 Y 𝑐 𝑎 𝑐 𝑑 𝑐 𝑠 2 𝑐 𝑑 ′ 1 𝛾 𝑠 2 ,𝑑 ′ 1 u 𝑐 𝑥 𝑝 𝑐 𝑢 1 𝑐 𝑤 𝑐 𝑤 ′ 𝛾 𝑥 𝑝 ,𝑢 1 C c 𝛾 𝑢 1 ,𝑤 2 R 𝑐 𝑎 𝑐 𝑧 𝑐 𝑠 3 𝑐 𝑑 ′ 1 𝛾 𝑧,𝑠 3 I i 𝑐 𝑥 𝑝 𝑐 𝑎 1 𝑐 𝑎 ′ 1 𝑐 𝑤 ′ 𝛾 𝑥 𝑝 ,𝑎 1 9 Y 𝑐 𝑎 𝑐 𝑧 𝑐 𝑎 ′ 1 𝑐 𝑤 ′ 𝛾 𝑎 ′ 1 ,𝑤 ′ H h 𝛾 𝑧,𝑎 ′ 1 i
In order to facilitate the reading of the generating triple confluence, we have omitted the context of the 2-cells 𝛾. Note also that some super columns may be empty and thus the indicated 2-cells 𝛾 may be identities.

The 3-sphere Ω 𝑥 𝑝 ,𝑢 The reduction 𝑅 Γ 3 induced by the order ⊳ on the 2-cells of Col 2 (S) and the order ◁ on 3-cells can be decomposed as follows. For all 𝑥 in S and 𝑣, 𝑡 in col(S) such that 𝑥 𝑣 𝑡 ×1 ×1 , the 2-cells 𝛾 𝑥,𝑣 , 𝛾 𝑣,𝑡 and 𝛾 𝑥,𝑣𝑡 are smaller then 𝛾 𝑥 𝑣,𝑡 for the order ⊳. Then the reduction 𝑅 Γ 3 removes the 2-cell 𝛾 𝑥 𝑣,𝑡 together with the 3-cell A 𝑥,𝑣,𝑡 . By iterating this reduction on the length of 𝑣, we reduce all the 2-cells of Col 2 (S) to the following set of 2-cells

{ 𝛾 𝑢,𝑣 | |𝑢 | ⩾ 1, |𝑣 | ⩾ 2 and 𝑢 𝑣 ×2 } ∪ { 𝛾 𝑢,𝑣 | |𝑢 | = 1, |𝑣 | ⩾ 1 and 𝑢 𝑣 ×1 }. (5)
For all 𝑥 in S and 𝑣, 𝑡 in col(S) such that 𝑥 𝑣 𝑡 ×1 ×2 with 𝑥𝑣 𝑡 × , the 2-cells 𝛾 𝑥,𝑣 , 𝛾 𝑣,𝑡 , 𝛾 𝑥,𝑤 and 𝛾 𝑎 ′ ,𝑤 ′ are smaller than 𝛾 𝑥 𝑣,𝑡 for the order ⊳. Then, the reduction 𝑅 Γ 3 removes the 2-cell 𝛾 𝑥 𝑣,𝑡 together with the 3-cell C 𝑥,𝑣,𝑡 . By iterating this reduction on the length of 𝑣, we reduce the set ( 5) to the following one:

{ 𝛾 𝑢,𝑣 | |𝑢 | = 1, |𝑣 | ⩾ 2 and 𝑢 𝑣 ×2 } ∪ { 𝛾 𝑢,𝑣 | |𝑢 | = 1, |𝑣 | ⩾ 1 and 𝑢 𝑣 ×1 }. (6)
For all 𝑥 in S and 𝑣, 𝑡 in col(S) such that 𝑥 𝑣 𝑡 ×2 ×1 , consider the following 3-cell:

𝑐 𝑒 𝑐 𝑒 ′𝑐 𝑡 𝑐 𝑒 𝛾 𝑒 ′ ,𝑡 7 W B 𝑥,𝑣,𝑡 Õ % 𝑐 𝑒 𝑐 𝑒 ′ 𝑡 𝛾 𝑒,𝑒 ′ 𝑡 $ D 𝑐 𝑥 𝑐 𝑣 𝑐 𝑡 𝛾 𝑥,𝑣 𝑐 𝑡 @ `𝑐𝑥 𝛾 𝑣,𝑡 7 W 𝑐 𝑠 𝑐 𝑠 ′ 𝑐 𝑥 𝑐 𝑣𝑡 𝛾 𝑥,𝑣𝑡 C c
where 𝛾 𝑒,𝑒 ′ 𝑡 is the 2-cell in (6) obtained from the 2-cell 𝛾 𝑒,𝑒 ′ 𝑡 by the previous step of the homotopical reduction by the 3-cell C 𝑥,𝑣,𝑡 . Since 𝑥 is in S, we have also that 𝑒 ′ is in S by definition of 𝛾. The 2-cells 𝛾 𝑥,𝑣 , 𝛾 𝑒 ′ ,𝑡 , 𝛾 𝑣,𝑡 and 𝛾 𝑒,𝑒 ′ 𝑡 being smaller than 𝛾 𝑥,𝑣𝑡 for the order ⊳, we can remove the 2-cells 𝛾 𝑥,𝑣𝑡 together with the 3-cell B 𝑥,𝑣,𝑡 . By iterating this reduction on the length of 𝑡, we reduce the set (6) to the following set:

{ 𝛾 𝑢,𝑣 | |𝑢 | = 1, |𝑣 | = 2 and 𝑢 𝑣 ×2 } ∪ { 𝛾 𝑢,𝑣 | |𝑢 | = 1, |𝑣 | ⩾ 1 and 𝑢 𝑣 ×1 }. (7)
Let us show that the set ( 7) is equal to the set of 2-cells PreCol 2 (S) defined in Subsection 2.9. It is sufficient to prove that PC 2 (S) is equal to the set of 2-cells 𝛾 𝑢,𝑣 in Col 2 (S) We extend𝑇 -1 𝜂,𝜀←𝛾 ′ into a Tietze transformation between the (3, 1)-polygraphs CPC 3 (S) ⊤ and Knuth cc 3 (S) ⊤ . Since CPC 3 (S) is a coherent presentation of P(S), we deduce the following result: Finally, we perform the homotopical reduction 𝑅 Γ 2 on the (3, 1)-polygraph Knuth cc 3 (S) whose collapsible part Γ 2 is defined by the 2-cells 𝜇 𝑢 of C 2 (S) and the order ≼ deglex . Thus, for every 2-cell 𝜇 𝑥 𝑝 ...𝑥 1 : 𝑐 𝑥 𝑝 . . . 𝑐 𝑥 1 ⇒ 𝑐 𝑥 𝑝 ...𝑥 1 in C 2 (S), we eliminate 𝑐 𝑥 𝑝 ...𝑥 1 together with 𝜇 𝑥 𝑝 ...𝑥 1 , from the bigger column to the smaller one with respect to the order ≼ deglex .

such that |𝑢 | = 1, |𝑣 | = 2 and 𝑢 𝑣 ×2 . Suppose that 𝑣 = 𝑥𝑥 ′ with 𝑥 ⩾ 𝑥 ′ in S with 𝑥 = 𝑥 ′ only if ||𝑥 || = 1. Since 𝑢 𝑣 ×2 , we obtain that 𝑢 ⩽ 𝑥 with 𝑢 = 𝑥 only if ||𝑢 || = 0. If 𝑢 ⩽ 𝑥 ′ with 𝑢 = 𝑥 ′ only if ||𝑢 || = 0, then 𝑅 𝑐𝑜𝑙 (⟦𝑢𝑣⟧ 𝑟 ) = (𝑥𝑢)𝑥 ′ and the 2-cell 𝛾 𝑢,𝑣 is equal to the 2-cell 𝛾 ′ 𝑢,𝑥𝑥 ′ : 𝑐 𝑢 𝑐 𝑥𝑥 ′ ⇒ 𝑐 𝑥𝑢 𝑐 𝑥 ′ . If 𝑥 ′ ⩽ 𝑢 with 𝑢 = 𝑥 ′ only if ||𝑢 || = 1, then 𝑅 𝑐𝑜𝑙 (⟦𝑢𝑣⟧ 𝑟 ) = (𝑢𝑥 ′ )𝑥,
Consider the following composite of Tietze transformations R := 𝑅 Γ 2 • 𝑇 -1 𝜂,𝜀←𝛾 ′ • 𝑇 -1 𝜇←𝛾 ′ • 𝑅 Γ 3 , defined from Col 3 (S) ⊤ to Knuth cc 3 (S) ⊤ as follows. Firstly, the transformation R eliminates the 3-cells of Col 3 (S) of the form A 𝑥,𝑣,𝑡 , B 𝑥,𝑣,𝑡 and C 𝑥,𝑣,𝑡 and reduces its set of 2-cells to PreCol 2 (S). Secondly, this transformation coherently replaces the 2-cells 𝜇 𝑥 𝑝 ...𝑥 1 by the 2-cells 𝛾 ′ 𝑥 𝑝 ,𝑥 𝑝-1 ...

=

Let M be a monoid and let Σ be an extended presentation of M. The (3, 1)-polygraph Σ is a coherent presentation of M if, and only if, for every 2-category C, there is an equivalence of categories between Act(M) and 2Cat(Σ * 1 /Σ 2 , C), that is natural in C, [START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF]. In this way, up to equivalence, the actions of a monoid M on categories are the same as the 2-functors from Σ * 1 /Σ 2 to Cat. As a consequence, Theorem 3.12 allows us to present actions of super plactic monoids on categories as follows:

3.14. Theorem. The category Act(P(S)) of actions of the monoid P(S) on categories is equivalent to the category of 2-functors from the free (2, 1)-category Knuth 2 (S) ⊤ generated by the 2-polygraph Knuth 2 (S) to the category Cat of categories, sending the 3-cells of Knuth 3 (S) to commutative diagrams in Cat.

  Note that when S = S 0 = [𝑛], the right (resp. left) insertion corresponds to the Schensted's right (resp. left) insertion introduced in[START_REF] Schensted | Longest increasing and decreasing subsequences[END_REF] on row-strict semistandard tableaux over [𝑛]. For any word 𝑤 = 𝑥 1 . . . 𝑥 𝑘 over S, denote by ⟦𝑤⟧ 𝑟 the super tableau obtained from 𝑤 by inserting its letters iteratively from left to right using the right insertion starting from the empty tableau:

Figure 1 : confluence diagrams 3 . 4 .

 134 Figure 1: confluence diagrams

3 . 6 .

 36 ′ ,𝑡 . Denote by 𝑑 1 and 𝑑 ′ 1 the readings of the two columns of ⟦𝑠 ′ 𝑡⟧ 𝑟 . The 3-cell X 𝑦,𝑠 ′ ,𝑡 being confluent and 𝑅 𝑐𝑜𝑙 (⟦𝑒 ′ 𝑡⟧ 𝑟 ) = 𝑏𝑏 ′ , we have 𝑅 𝑐𝑜𝑙 (⟦𝑦𝑑 1 ⟧ 𝑟 ) = 𝑏𝑠 2 and 𝑅 𝑐𝑜𝑙 (⟦𝑠 2 𝑑 ′ 1 ⟧ 𝑟 ) = 𝑏 ′ . On the other hand, the 3-cell X 𝑢 1 ,𝑣,𝑡 being confluent, we have 𝑅 𝑐𝑜𝑙 (⟦𝑠𝑑1 ⟧ 𝑟 ) = 𝑎 1 𝑠 3 and 𝑅 𝑐𝑜𝑙 (⟦𝑎 ′ 1 𝑤 ′ ⟧ 𝑟 ) = 𝑠 3 𝑑 ′ 1 . Finally, since X 𝑥 𝑝 ,𝑠,𝑑 1 is confluent, we deduce that 𝑅 𝑐𝑜𝑙 (⟦𝑧𝑠 3 ⟧ 𝑟 ) = 𝑑𝑠 2 .Denote by Col 3 (S) the extended presentation of the monoid P(S) obtained from Col 2 (S) by adjunction of one family of 3-cells X 𝑥,𝑣,𝑡 of the form (4), for all 𝑥 in S and 𝑣, 𝑡 in col(S) such that 𝑥 𝑣 𝑡 × × .Let Γ 4 be the collapsible part made of the family of 3-sphere Ω 𝑥 𝑝 ,𝑢 1 ,𝑣,𝑡 , indexed by 𝑥 𝑝 in S and 𝑢 1 , 𝑣, 𝑡 in col(S) such that 𝑢 𝑣 𝑡 × × and 𝑢 = 𝑥 𝑝 𝑢 1 . We consider the order ◁ on the 3-cells of Col 3 (S) defined by A 𝑢,𝑣,𝑡 ◁ C ′ 𝑢,𝑣,𝑡 ◁ C 𝑢,𝑣,𝑡 ◁ B 𝑢,𝑣,𝑡 ◁ D 𝑢,𝑣,𝑡 , and if X 𝑢,𝑣,𝑡 ∈ {A 𝑢,𝑣,𝑡 , B 𝑢,𝑣,𝑡 , C 𝑢,𝑣,𝑡 , C ′ 𝑢,𝑣,𝑡 , D 𝑢,𝑣,𝑡 } with 𝑢 ′ ≼ deglex 𝑢, then X 𝑢 ′ ,𝑣 ′ ,𝑡 ′ ◁ X 𝑢,𝑣,𝑡 , for all 𝑢, 𝑣, 𝑡 in col(S) such that 𝑢 𝑣 𝑡 × × . By construction of the 3sphere Ω 𝑥 𝑝 ,𝑢 1 ,𝑣,𝑡 , its source contains the 3-cell X 𝑢 1 ,𝑣,𝑡 and its target contains the 3-cell X 𝑢,𝑣,𝑡 with |𝑢 1 | < |𝑢 |. Up to a Nielsen transformation, by applying the reduction 𝑅 Γ 4 on the (3, 1)-polygraph Col 3 (S) with respect to Γ 4 and the order ◁, we obtain the (3, 1)-polygraph Col 3 (S). Hence, we deduce the following result. Proposition. The (3, 1)-polygraph Col 3 (S) is a coherent presentation of the monoid P(S).

3. 7 . 3 .

 73 Super pre-column coherent presentation. We reduce the coherent presentation Col 3 (S) into a coherent presentation whose underlying 2-polygraph is PreCol 2 (S). This reduction is obtained using the homotopical reduction 𝑅 Γ 3 on the (3, 1)-polygraph Col 3 (S) whose collapsible part Γ 3 is the set of 3-cells A 𝑥,𝑣,𝑡 , B 𝑥,𝑣,𝑡 and C 𝑥,𝑣,𝑡 with 𝑥 ∈ S and 𝑣, 𝑡 ∈ col(S) such that 𝑥 𝑣 𝑡 ×1 ×1 , 𝑥 𝑣 𝑡 ×2 ×1 and 𝑥 𝑣 𝑡 ×1 ×2 Coherent presentations of the super plactic monoid with 𝑥𝑣 𝑡 × respectively, and the order ⊳ on the 2-cells of Col 2 (S) defined by 𝛾 𝑢 ′ ,𝑣 ′ ⊳ 𝛾 𝑢,𝑣 if |𝑢𝑣 | > |𝑢 ′ 𝑣 ′ |, or |𝑢𝑣 | = |𝑢 ′ 𝑣 ′ | and |𝑢 | > |𝑅 𝑐𝑜𝑙 (⟦𝑢 ′ 𝑣 ′ ⟧ 𝑟 )| or |𝑢 | ⩽ |𝑅 𝑐𝑜𝑙 (⟦𝑢 ′ 𝑣 ′ ⟧ 𝑟 )| and 𝑢 ′ ≼ rev 𝑢, for all 𝑢, 𝑣, 𝑢 ′ , 𝑣 ′ ∈ col(S) such that 𝑢 𝑣 × and 𝑢 ′ 𝑣 ′ × .

3. 11 .

 11 Lemma. The (3,1)-polygraph Knuth cc 3 (S) is a coherent presentation of the monoid P(S).

𝑥 1 ,

 1 for each column 𝑥 𝑝 . . . 𝑥 1 such that |𝑥 𝑝 . . . 𝑥 1 | > 2, the 2-cells 𝛾 ′ 𝑥,𝑧𝑦 by 𝜂 𝑐 𝑥,𝑦,𝑧 , for any 𝑥 ⩽ 𝑦 ⩽ 𝑧, with 𝑥 = 𝑦 only if ||𝑦|| = 0 and 𝑦 = 𝑧 only if ||𝑦|| = 1 and the 2-cells 𝛾 ′ 𝑦,𝑧𝑥 by 𝜀 𝑐 𝑥,𝑦,𝑧 for any 𝑥 ⩽ 𝑦 ⩽ 𝑧, with 𝑥 = 𝑦 only if ||𝑦|| = 1 and 𝑦 = 𝑧 only if ||𝑦|| = 0. Finally, for each column 𝑢, the transformation R eliminates the generator 𝑐 𝑢 together with the 2-cell 𝜇 𝑢 with respect to the order ≼ deglex . Let Knuth 3 (S) be the extended presentation of the monoid P(S) obtained from Knuth 2 (S) by adjunction of the 3-cells R (C ′ 𝑥,𝑣,𝑡 ) for 𝑥 𝑣 𝑡 ×1 ×2 with 𝑥𝑣 𝑡 and R (D 𝑥,𝑣,𝑡 ) for 𝑥 𝑣 𝑡 ×2 ×2 . The transformation R being a composite of Tietze transformations, we deduce the following result.3.12. Theorem. The (3, 1)-polygraph Knuth 3 (S) is a coherent presentation of the monoid P(S).

3. 13 .

 13 Actions of super plactic monoids on categories. A monoid M can be seen as a 2-category with exactly one 0-cell •, with the elements of the monoid M as 1-cells and with identity 2-cells only. The category of actions of M on categories is the category Act(M) of 2-representations of M in the category Cat of categories. The full subcategory of Act(M) whose objects are the 2-functors is denoted by 2Cat(M, Cat). We refer the reader to[START_REF] Gaussent | Coherent presentations of Artin monoids[END_REF] for a full introduction on the category of 2-representations of 2-categories. More explicitly, an action 𝐴 of the monoid M is specified by a category C = 𝐴(•), an endofunctor 𝐴(𝑢) : C → C for every 𝑢 in M, a natural isomorphism 𝐴 𝑢,𝑣 : 𝐴(𝑢)𝐴(𝑣) ⇒ 𝐴(𝑢𝑣) for every elements 𝑢 and 𝑣 of M, and a natural isomorphism 𝐴 • : 1 C ⇒ 𝐴(1) such that: i) for every triple (𝑢, 𝑣, 𝑤) of elements of the monoid M, the following diagram commutes 𝐴(𝑢𝑣)𝐴(𝑤) 𝐴 𝑢𝑣,𝑤 ' G = 𝐴(𝑢)𝐴(𝑣)𝐴(𝑤) 𝐴 𝑢,𝑣 𝐴(𝑤) 9 Y 𝐴(𝑢)𝐴 𝑣,𝑤 5 U 𝐴(𝑢𝑣𝑤) 𝐴(𝑢)𝐴(𝑣𝑤) 𝐴 𝑢,𝑣𝑤 G g ii) for every element 𝑢 of the monoid M, the following diagrams commute 𝐴(1)𝐴(𝑢) 𝐴 1,𝑢

  𝑐 𝑎 𝑐 𝑑 𝑐 𝑏 ′ 𝑐 𝑢 𝑐 𝑤 𝑐 𝑤 ′ 𝛾 𝑢,𝑤 𝑐 𝑤 ′ 7 W 𝑐 𝑎 𝑐 𝑎 ′𝑐 𝑤 ′ 𝑐 𝑎 𝛾 𝑎 ′ ,𝑤

7 W X 𝑢,𝑣,𝑡 Õ % 𝑐 𝑒 𝑐 𝑏 𝑐 𝑏 ′ 𝛾 𝑒,𝑏 𝑐 𝑏 ′ ' G 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 𝑐 𝑡 9 Y 𝑐 𝑢 𝛾 𝑣,𝑡 5 U ′ F f for every 𝑐 𝑢 , 𝑐 𝑣 and 𝑐 𝑡 in Col 1 (S) such that the juxtapositions of ⟦𝑢⟧ 𝑟 and ⟦𝑣⟧ 𝑟 , and of ⟦𝑣⟧ 𝑟 and ⟦𝑡⟧ 𝑟 do not form super tableaux. Following Squier's coherence theorem, [33], Theorem 3.3 states that Col 3 (S)

  1 . . . 𝑥 𝑘 a word in A * of length 𝑘, where 𝑥 1 , . . . , 𝑥 𝑘 belong to A. The length of a word 𝑤 will be denoted by |𝑤 |. Let 𝑤 = 𝑥 1 . . . 𝑥 𝑘 be a word in A >0 . Let S be a finite or countable totally ordered set and ||.|| : S → Z 2 be any map, where Z 2 = {0, 1} denotes the additive cyclic group of order 2. The ordered pair (S, ||.||) is called a signed alphabet, and we denote S 0 = {𝑎 ∈ S ||𝑎|| = 0} and S 1 = {𝑎 ∈ S ||𝑎|| = 1}. A monoid M is said a Z 2 -graded monoid or a supermonoid if a map ||.|| : M → Z 2 is given such that ||𝑢.𝑣 || = ||𝑢 || + ||𝑣 ||, for all 𝑢 and 𝑣 in M. We call ||𝑢 || the Z 2 -degree of the element 𝑢. The free monoid S * over S is Z 2 -graded by considering ||𝑤 || := ||𝑥 1 || + . . . + ||𝑥 𝑘 ||, for any word 𝑤 = 𝑥 1 . . . 𝑥 𝑘 in S * . In the rest of this article, and if there is no possible confusion, S denotes a signed alphabet.

* . We denote by ℓ (𝑤) the leftmost letter of 𝑤 and by Rem(𝑤) the subword of 𝑤 such that 𝑤 = ℓ (𝑤) Rem(𝑤). A word 𝑤 ′ is a subsequence of 𝑤 if 𝑤 ′ = 𝑥 𝑖 1 . . . 𝑥 𝑖 𝑙 with 1 ⩽ 𝑖 1 < . . . < 𝑖 𝑙 < 𝑘. We will denote by [𝑛] the ordered set {1 < . . . < 𝑛} for 𝑛 in Z

  the Nielsen transformation 𝜅 𝜇←𝛼 is the Tietze transformation that replaces in Σ ⊤ 2 the 2-cell 𝜇 by a 2-cell 𝛼 : 𝑠 1 (𝜇 1 ) ⇒ 𝑡 1 (𝜇 2 ). When 𝜇 2 is identity, we denote by 𝜅 𝜇←𝛼 the Nielsen transformation which, given a 2-cell 𝜇 1 ★ 1 𝜇 in Σ ⊤ 2 , replaces the 2-cell 𝜇 by a 2-cell 𝛼 : 𝑠 1 (𝜇 1 ) ⇒ 𝑡 1 (𝜇).2.2. Super Young tableaux and insertionsA rewriting step of a 2-polygraph Σ is a 2-cell of Σ * 2 with shape 𝑤𝛼𝑤 ′ ,where 𝛼 is a 2-cell of Σ 2 and 𝑤 and 𝑤 ′ are 1-cells of Σ * 1 . A rewriting sequence of Σ is a finite or infinite sequence of rewriting steps. A 1-cell 𝑢 of Σ * 1 is a normal form if there is no rewriting step with source 𝑢. The 2-polygraph Σ terminates if it has no infinite rewriting sequence. A branching of Σ is a non ordered pair (𝑓 , 𝑔) of 2-cells of Σ * 2 such that 𝑠 1 (𝑓 ) = 𝑠 1 (𝑔). A branching (𝑓 , 𝑔) is local if 𝑓 and 𝑔 are rewriting steps. A branching is aspherical if it is of the form (𝑓 , 𝑓 ), for a rewriting step 𝑓 and Peiffer when it is of the form (𝑓 𝑣, 𝑢𝑔) for rewriting steps 𝑓 and 𝑔 with 𝑠 1 (𝑓 ) = 𝑢 and 𝑠 1 (𝑔) = 𝑣. The overlapping branchings are the remaining local branchings. An overlapping local branching is critical if it is minimal for the order ⊑ generated by the relations (𝑓 , 𝑔) ⊑ 𝑤 𝑓 𝑤 ′ , 𝑤𝑔𝑤 ′ ), given for any local branching (𝑓 , 𝑔) and any possible 1-cells 𝑤 and 𝑤 ′ of the category Σ * 1 . A branching (𝑓 , 𝑔) is confluent if there exist 2-cells 𝑓 ′ and 𝑔 ′ in Σ * 2 such that 𝑠 1

′

  Proof. Consider 𝑢 = 𝑥 1 . . . 𝑥 𝑝 and 𝑣 = 𝑦 1 . . . 𝑦 𝑞 in col(S) such that 𝑢 𝑣 × . If 𝑥 𝑝 ⩾ 𝑦 1 with 𝑥 𝑝 = 𝑦 1 only if ||𝑥 𝑝 || = 1, then the super tableau ⟦𝑢𝑣⟧ 𝑟 consists of only column whose reading is 𝑢𝑣. Otherwise, suppose 𝑥 𝑝 ⩽ 𝑦 1 with 𝑥 𝑝 = 𝑦 1 only if ||𝑥 𝑝 || = 0. Since the words 𝑢 and 𝑣 are decreasing with respect to S 1 , the longest super row word that is a subsequence of 𝑢𝑣 contains one element from each of 𝑢 and 𝑣. Then its length is equal to 2, showing by Lemma 2.4 that ⟦𝑢𝑣⟧ 𝑟 contains two columns. Suppose now that ⟦𝑢𝑣⟧ 𝑟 contains exactly two columns. Then its leftmost column is obtained by inserting some elements of 𝑣 into 𝑢, and by keeping the elements of 𝑢 unchangeable, showing that it contains more elements than 𝑢. Super column presentation of the super plactic monoid 2.7. Super columns as generators. Suppose S = {𝑥 1 , 𝑥 2 , . . .}. Let Col 1 (S) = 𝑐 𝑢 𝑢 ∈ col(S) be the set of super column generators of the super plactic monoid P(S) and C 2 (S) = 𝜇 𝑢 : 𝑐 𝑥 𝑝 . . . 𝑐 𝑥 1 ⇒ 𝑐 𝑢 𝑢 = 𝑥 𝑝 . . . 𝑥 1 ∈ col(S) with |𝑢 | ⩾ 2 be the set of the defining relations for the super column generators. We will denote by Knuth c 2 (S) the 2-polygraph whose set of 1-cells is {𝑐 𝑥 1 , 𝑐 𝑥 2 , . . .} and whose 2-cells are 𝜂 𝑐 𝑥,𝑦,𝑧 : 𝑐 𝑧 𝑐 𝑥 𝑐 𝑦 ⇒ 𝑐 𝑥 𝑐 𝑧 𝑐 𝑦 , with 𝑥 = 𝑦 only if ||𝑦|| = 0 and 𝑦 = 𝑧 only if ||𝑦|| = 1, 𝜀 𝑐 𝑥,𝑦,𝑧 : 𝑐 𝑦 𝑐 𝑧 𝑐 𝑥 ⇒ 𝑐 𝑦 𝑐 𝑥 𝑐 𝑧 , with 𝑥 = 𝑦 only if ||𝑦|| = 1 and 𝑦 = 𝑧 only if ||𝑦|| = 0, for any 𝑥 ⩽ 𝑦 ⩽ 𝑧 of elements of S. By definition, this 2-polygraph is Tietze equivalent to Knuth 2 (S). In the sequel, if there is no possible confusion, we will identify the 2-polygraphs Knuth c 2 (S) and Knuth 2 (S). Denote by Knuth cc 2 (S) the 2-polygraph whose set of 1-cells is {𝑐 𝑥 1 , 𝑐 𝑥 2 , . . .} and whose set of 2-cells is C 2 (S) ∪ Knuth c 2 (S). 𝑥 𝑝 . . . 𝑥 1 in col(S) such that |𝑢 | ⩾ 2, and the corresponding collapsible 2-cell 𝜇 𝑢 : 𝑐 𝑥 𝑝 . . . 𝑐 𝑥 1 ⇒ 𝑐 𝑢 , we obtain that the 2-polygraphs Knuth c The 2-polygraph PreCol 2 (S) is a presentation of the super plactic monoid P(S). . We first prove that 2-polygraph CPC 2 (S) whose set of 1-cells is Col 1 (S) and set of 2-cells is C 2 (S) ∪ PC 2 (S), is Tietze equivalent to Knuth cc 2 (S). We consider the following critical branching 𝑐 𝑥 𝑐 𝑧 𝑐 𝑦 𝑐 𝑥 𝜇 𝑧𝑦 7 W 𝑐 𝑥 𝑐 𝑧𝑦 𝑐 𝑧 𝑐 𝑥 𝑐 𝑦 𝜇 𝑧𝑥 𝑐 𝑦 7 W 𝑐 𝑧𝑥 𝑐 𝑦 of the 2-polygraph Knuth cc 2 (S), for any 𝑥 ⩽ 𝑦 ⩽ 𝑧 with 𝑥 = 𝑦 only if ||𝑦|| = 0 and 𝑦 = 𝑧 only if ||𝑦|| = 1, and the Tietze transformation 𝜅 𝜂 𝑐 𝑥,𝑦,𝑧 ←𝛾 ′ 𝑥,𝑧𝑦 for the 2-cell 𝜂 𝑐 𝑥,𝑦,𝑧 , and denote by 𝑇 𝜂←𝛾 ′ the successive applications of 𝜅 𝜂 𝑐 𝑥,𝑦,𝑧 ←𝛾 ′ 𝑥,𝑧𝑦 , for any 𝑥 ⩽ 𝑦 ⩽ 𝑧 with 𝑥 = 𝑦 only if ||𝑦|| = 0 and 𝑦 = 𝑧 only if ||𝑦|| = 1, with respect to the lexicographic order on the triples (𝑥, 𝑦, 𝑧) induced by the total order on S. Similarly, we study the critical branching (𝜀 𝑐 𝑥,𝑦,𝑧 , 𝑐 𝑦 𝜇 𝑧𝑥 ) of Knuth cc 2 (S), for any 𝑥 ⩽ 𝑦 ⩽ 𝑧 with 𝑥 = 𝑦 only if ||𝑦|| = 1 and 𝑦 = 𝑧 only if ||𝑦|| = 0, by introducing the Tietze transformation 𝜅 𝜀 𝑐 𝑥,𝑦,𝑧 ←𝛾 ′ 𝑦,𝑧𝑥 from Knuth cc 2 (S) ⊤ 2.11. Super column presentation to Knuth cc 2 (S) ⊤ /(𝜀 𝑐 𝑥,𝑦,𝑧 ← 𝛾 ′ 𝑦,𝑧𝑥 ). Denote by 𝑇 𝜀←𝛾 ′ the successive applications of this Tietze transformation with respect to the lexicographic order on the triples (𝑥, 𝑦, 𝑧) induced by the total order on S, and by 𝑇 𝜂,𝜀←𝛾 ′ the Tietze transformation from Knuth cc 2 (S) ⊤ to CPC 2 (S) ⊤ given by 𝑇 𝜂←𝛾 ′ • 𝑇 𝜀←𝛾 ′ . Finally, we prove that the 2-polygraphs CPC 2 (S) and PreCol 2 (S) are Tietze equivalents. Let 𝑥 𝑝 . . . 𝑥 1 be in col(S) with |𝑥 𝑝 . . . 𝑥 1 | > 2 and define 𝛾 ′ 𝑦,𝑥 := 𝜇 𝑦𝑥 : 𝑐 𝑦 𝑐 𝑥 ⇒ 𝑐 𝑦𝑥 , for any 𝑥 ⩽ 𝑦 with 𝑥 = 𝑦 only if ||𝑥 || = 1. We consider the following critical branching 𝑐 𝑥 𝑝 𝑐 𝑥 𝑝-1 ...𝑥 1 𝑐 𝑥 𝑝 . . . 𝑐 𝑥 1 𝑐 𝑥 𝑝 𝜇 𝑥 𝑝-1 ...𝑥 1 A a 𝜇 𝑥 𝑝 ...𝑥 1 7 W 𝑐 𝑥 𝑝 ...𝑥 1 of the 2-polygraph CPC 2 (S) and the following Tietze transformation 𝜅 CPC 2 (S) ⊤ /(𝜇 𝑥 𝑝 ...𝑥 1 ← 𝛾 ′ 𝑥 𝑝 ,𝑥 𝑝-1 ...𝑥 1 ), that substitutes the 2-cell 𝛾 ′ 𝑥 𝑝 ,𝑥 𝑝-1 ...𝑥 1 for the 2-cell 𝜇 𝑥 𝑝 ...𝑥 1 , for each column 𝑥 𝑝 . . . 𝑥 1 such that 𝑝 > 2. Starting from CPC 2 (S), we apply successively the Tietze transformation 𝜅

	Proof𝜂 𝑐 𝑥,𝑦,𝑧	B b
	that substitutes the 2-cell 𝛾 ′	𝑥,𝑧𝑦 : Knuth cc 2 (S) ⊤ -→ Knuth cc 2 (S) ⊤ /(𝜂 𝑐 𝑥,𝑦,𝑧 ← 𝛾 ′ 𝑥,𝑧𝑦 ),

i) we will denote 𝑢 𝑣 if the topmost juxtaposition of ⟦𝑢⟧ 𝑟 and ⟦𝑣⟧ 𝑟 forms a super tableau, that is, |𝑢 | ⩾ |𝑣 | and 𝑥 𝑖 ⩽ 𝑦 𝑖 , for any 𝑖 ⩽ |𝑣 |, with 𝑥 𝑖 = 𝑦 𝑖 only if ||𝑥 𝑖 || = 0, ii) we will denote 𝑢 𝑣 × in all the other cases, that is, when |𝑢 | < |𝑣 | or 𝑥 𝑖 ⩾ 𝑦 𝑖 , for some 𝑖 ⩽ |𝑣 |, with 𝑥 𝑖 = 𝑦 𝑖 only if ||𝑥 𝑖 || = 1. 2.6. Lemma. Let 𝑢 and 𝑣 be in col(S) such that 𝑢 𝑣 × . The super tableau ⟦𝑢𝑣⟧ 𝑟 contains at most two columns. Moreover, if ⟦𝑢𝑣⟧ 𝑟 contains exactly two columns, the left column contains more elements than 𝑢. □ When 𝑢 𝑣 × , we will denote 𝑢 𝑣 ×1 if the super tableau ⟦𝑢𝑣⟧ 𝑟 consists of one column and by 𝑢 𝑣 ×2 if it consists of two columns. 2. 2.8. Proposition. The 2-polygraph Knuth cc 2 (S) is a presentation of the super plactic monoid P(S). Proof. By adding to the 2-polygraph Knuth c 2 (S) all the column generators 𝑐 𝑢 , for all 𝑢 = 2 (S) and Knuth cc 2 (S) are Tietze-equivalent, showing the claim. □ 2.9. Super pre-column presentation. Denote by PC 2 (S) the set of 2-cells of the following form: 𝛾 ′ 𝑥,𝑧𝑦 : 𝑐 𝑥 𝑐 𝑧𝑦 ⇒ 𝑐 𝑧𝑥 𝑐 𝑦 , with 𝑥 = 𝑦 only if ||𝑦|| = 0 and 𝑦 = 𝑧 only if ||𝑦|| = 1, 𝛾 ′ 𝑦,𝑧𝑥 : 𝑐 𝑦 𝑐 𝑧𝑥 ⇒ 𝑐 𝑦𝑥 𝑐 𝑧 , with 𝑥 = 𝑦 only if ||𝑦|| = 1 and 𝑦 = 𝑧 only if ||𝑦|| = 0, for any 𝑥 ⩽ 𝑦 ⩽ 𝑧 elements of S. Define the 2-polygraph PreCol 2 (S) whose set of 1-cells is Col 1 (S) and whose set of 2-cells is PC 2 (S) ∪ 𝛾 ′ 𝑥,𝑢 : 𝑐 𝑥 𝑐 𝑢 ⇒ 𝑐 𝑥𝑢 | 𝑥𝑢 ∈ col(S) and 𝑥 ∈ S . 2.10. Proposition. ′ 𝜇 𝑥𝑝 ...𝑥 1 ←𝛾 ′ 𝑥𝑝 ,𝑥 𝑝-1 ...𝑥 1 : CPC 2 (S) ⊤ -→ ′ 𝜇 𝑥𝑝 ...𝑥 1 ←𝛾 ′ 𝑥𝑝 ,𝑥 𝑝-1 ...𝑥 1

  1, defines a Tietze transformation from CPC 2 (S) ⊤ to PreCol 2 (S) ⊤ . The rest of this section is devoted to prove this result. We first show in Lemma 2.13 that the 2polygraph Col 2 (S) is terminating and confluent. We then show in Lemma 2.14 that it is a presentation of the super plactic monoid by showing that it is Tietze equivalent to the 2-polygraph Knuth cc 2 (S). 2.13. Lemma. The 2-polygraph Col 2 (S) is convergent. Proof. The termination of Col 2 (S) is proved using the length-lexicographic order ≪ on Col 1 (S) * with respect to ≼ rev defined by setting 𝑐 𝑢 1 . . . 𝑐 𝑢 𝑘 ≪ 𝑐 𝑣 1 . . . 𝑐 𝑣 𝑙 , if 𝑘 < 𝑙 or 𝑘 = 𝑙 and there exists 𝑖 ∈ {1, . . . , 𝑘 } such that 𝑢 𝑖 ≼ rev 𝑣 𝑖 and 𝑐 𝑢 𝑗 = 𝑐 𝑣 𝑗 , for any 𝑗 < 𝑖. We prove that ≪ is a well-ordering on Col 1 (S) * that is compatible with rules in Col 2 (S), that is if 𝑈 𝑐 𝑢 𝑐 𝑣 𝑉 ⇒ 𝑈 𝑐 𝑤 𝑐 𝑤 ′𝑉 then 𝑈 𝑐 𝑢 𝑐 𝑣 𝑉 ≪ 𝑈 𝑐 𝑤 𝑐 𝑤 ′𝑉 , for all 𝑈 𝑐 𝑢 𝑐 𝑣 𝑉 and 𝑈 𝑐 𝑤 𝑐 𝑤 ′𝑉 in Col 1 (S) * . If 𝑐 𝑤 ′ = 1, then 𝑈 𝑐 𝑢 𝑐 𝑣 𝑉 contains more elements than 𝑈 𝑐 𝑤 𝑉 , showing that 𝑈 𝑐 𝑤 𝑉 ≪ 𝑈 𝑐 𝑢 𝑐 𝑣 𝑉 . On the other hand, suppose that the super tableau ⟦𝑢𝑣⟧ 𝑟 consists of two columns whose readings of are denoted by 𝑤 and 𝑤 ′ respectively. By Lemma 2.6, the word 𝑤 contains more elements than 𝑢. Then, we obtain 𝑤 ≼ rev 𝑢, showing that 𝑈 𝑐 𝑤 𝑉 ≪ 𝑈 𝑐 𝑢 𝑐 𝑣 𝑉 . Since every application of a rule yield a ≪-preceding word, it follows that every sequence of rewriting using the 2-cells of Col 2 (S) must terminate. We show that the 2-polygraph Col 2 (S) is locally confluent and thus confluent by the termination hypothesis. Any critical branching of the 2-polygraph Col 2 (S) has the following form: 𝑐 𝑒 𝑐 𝑒 ′𝑐 𝑡 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 𝑐 𝑡 9 Y 𝑐 𝑢 𝛾 𝑣,𝑡 9 Y 𝑐 𝑢 𝑐 𝑤 𝑐 𝑤 ′ (3) for every 𝑢, 𝑣 and 𝑡 in Col(S) such that 𝑢 𝑣 𝑡 × × , where 𝑒 and 𝑒 ′ (resp. 𝑤 and 𝑤 ′ ) are the readings of the two columns of the super tableau ⟦𝑢𝑣⟧ 𝑟 (resp. ⟦𝑣𝑡⟧ 𝑟 ). Following the definition of the right insertion algorithm, and by applying the leftmost (resp. rightmost) rewriting path with respect to Col 2 (S) on 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 , we yield to the three super columns generators corresponding to the columns of the super tableau ⟦𝑢⟧ 𝑟 ★ 𝑟 ⟦𝑣⟧ 𝑟 ★ 𝑟 ⟦𝑡⟧ 𝑟 (resp. ⟦𝑢⟧ 𝑟 ★ 𝑟 ⟦𝑣⟧ 𝑟 ★ 𝑟 ⟦𝑡⟧ 𝑟 ). By the associativity of the product ★ 𝑟 , we have ⟦𝑢⟧ 𝑟 ★ 𝑟 ⟦𝑣⟧ 𝑟 ★ 𝑟 ⟦𝑡⟧ 𝑟 = ⟦𝑢⟧ 𝑟 ★ 𝑟 ⟦𝑣⟧ 𝑟 ★ 𝑟 ⟦𝑡⟧ 𝑟 , showing that the critical branching (3) is confluent. 𝑐 𝑦 𝑐 𝑧𝑥 𝛾 𝑦,𝑧𝑥 7 W 𝑐 𝑦𝑥 𝑐 𝑧 𝛾 𝑥 𝑝 ,𝑥 𝑝-1 𝑐 𝑥 𝑝-2 . . . 𝑐 𝑥 1 𝑐 𝑥 𝑝 𝑥 𝑝-1 𝑐 𝑥 𝑝-2 . . . 𝑐 𝑥 1 7 W (. . . ) 7 W 𝑐 𝑥 𝑝 ...𝑥 2 𝑐 𝑥 1 𝛾 𝑥 𝑝 ...𝑥 2 ,𝑥 1 As a consequence, if the words 𝑤 and 𝑤 ′ in Col 1 (S) * are equal modulo relations in Knuth cc 2 (S), then they are equal modulo relations in Col 2 (S). Conversely, following Subsection 2.3, the following equivalence 𝑢𝑣 ∼ P( S) 𝑅 𝑟𝑜𝑤 (⟦𝑢𝑣⟧ 𝑟 ) ∼ P( S) 𝑅 𝑐𝑜𝑙 (⟦𝑢𝑣⟧ 𝑟 ) holds, for all 𝑢, 𝑣 in Col(S) such that 𝑢 𝑣 × . In this way, if two words in Col 1 (S)

					𝑐 𝑦 𝑐 𝑧 𝑐 𝑥	𝜀 𝑐 𝑥,𝑦,𝑧 7 W 𝑐 𝑦 𝑐 𝑥 𝑐 𝑧
	𝑐 𝑧𝑥 𝑐 𝑦	𝛾 𝑥,𝑧𝑦 e y	Õ % 𝑐 𝑥 𝑐 𝑧𝑦	(resp.	𝑐 𝑦 𝛾 𝑧,𝑥	Õ %	𝛾 𝑦,𝑥 𝑐 𝑧 Õ %

□ 2.11. Super column presentation. Consider the 2-polygraph Col 2 (S) whose set of 1-cells is Col 1 (S) and whose 2-cells are 𝛾 𝑢,𝑣 : 𝑐 𝑢 𝑐 𝑣 ⇒ 𝑐 𝑤 𝑐 𝑤 ′ for every 𝑢 and 𝑣 in col(S) such that 𝑢 𝑣 × , where 𝑤 and 𝑤 ′ are respectively the readings of the left and right columns of the super tableau ⟦𝑢𝑣⟧ 𝑟 , if 𝑢 𝑣 ×2 , and 𝑤 = 𝑢𝑣 and 𝑐 𝑤 ′ = 1, if 𝑢 𝑣 ×1 . 2.12. Theorem. The 2-polygraph Col 2 (S) is a convergent presentation of the super plactic monoid P(S). □ 2.14. Lemma. The 2-polygraph Col 2 (S) is a presentation of the super plactic monoid P(S). Proof. Prove that the 2-polygraph Col 2 (S) is Tietze equivalent to the 2-polygraph Knuth cc 2 (S). Any 2-cell in Knuth cc 2 (S) can be deduced from a 2-cell in Col 2 (S) as follows. For any 𝑥 ⩽ 𝑦 ⩽ 𝑧 in S, with 𝑥 = 𝑦 only if ||𝑦|| = 0 and 𝑦 = 𝑧 only if ||𝑦|| = 1 (resp. 𝑥 = 𝑦 only if ||𝑦|| = 1 and 𝑦 = 𝑧 only if ||𝑦|| = 0), the 2-cells 𝜂 𝑐 𝑥,𝑦,𝑧 (resp. 𝜀 𝑐 𝑥,𝑦,𝑧 ) can be deduced by the following composition 𝑐 𝑧 𝑐 𝑥 𝑐 𝑦 𝜂 𝑐 𝑥,𝑦,𝑧 7 W 𝛾 𝑧,𝑥 𝑐 𝑦 Õ % 𝑐 𝑥 𝑐 𝑧 𝑐 𝑦 𝑐 𝑥 𝛾 𝑧,𝑦 ). Moreover, for any super column 𝑥 𝑝 . . . 𝑥 1 , the 2-cell 𝜇 𝑥 𝑝 ...𝑥 1 can be deduced by the following composition 𝑐 𝑥 𝑝 . . . 𝑐 𝑥 1 𝜇 𝑥 𝑝 ...𝑥 1 7 W Õ % 𝑐 𝑥 𝑝 ...𝑥 1 i * are equal modulo relations in Col 2 (S), they are super plactic congruent and hence they are equal modulo Knuth cc 2 (S), showing the claim. □

  3.1. Coherent presentations of monoids. A (3, 1)-polygraph is a pair (Σ 2 , Σ 3 ) made of a 2-polygraph Σ 2 and a globular extension Σ 3 of the (2, 1)-category Σ ⊤ 2 , that is a set of 3-cells 𝐴 : 𝑓 ⇛ 𝑔 relating 2-cells 𝑓 and 𝑔 in Σ ⊤ 2 , respectively denoted by 𝑠 2 (𝐴) and 𝑡 2 (𝐴) and satisfying the globular relations 𝑠 1 𝑠 2

  Hence, the generating syzygy of the extended presentation of the 2-polygraph Col 2 (S) has the following form𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝑐 𝑎 𝑐 𝑑 𝑐 𝑏 ′ Õ % for all 𝑐 𝑢 , 𝑐 𝑣 , 𝑐 𝑡 in Col 1 (S) such that 𝑢 𝑣 𝑡 × × ,where 𝑎, 𝑑, 𝑏 ′ are the readings of the three columns of the super tableau ⟦𝑢𝑣𝑡⟧ 𝑟 , and where the 2-source (resp. 2-target) of the syzygy corresponds to the application of the right (resp. left) insertion ∅ ⇝ 𝑢𝑣𝑡 (resp. (𝑢𝑣𝑡 ⇝ ∅)) on the word 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 . Finally, by definition of the right (resp. left) insertion algorithm ⇝ (resp. ⇝), we can lead to the normal form 𝑐 𝑎 𝑐 𝑑 𝑐 𝑏 ′ by applying at most three steps of reduction on the initial word 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 , showing the claim. □ 3.4. Homotopical reduction procedure 3.3.1. Remark. It is worth noting that there are fives forms for the generating syzygy (4) depending on the following four cases: 𝑢 𝑣 𝑡 ×1 ×1 , 𝑢 𝑣 𝑡 ×2 ×1 , 𝑢 𝑣 𝑡 ×1 ×2 and 𝑢 𝑣 𝑡 ×2 ×2 , as shown in Figure 1. These forms are obtained by a case-by-case analysis and they will be used in the sequel in order to reduce the super column coherent presentation Col 3 (S) into a smaller one over the initial signed alphabet. 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 𝑐 𝑡 9 Y 𝑐 𝑒 𝑐 𝑒 ′𝑐 𝑡 𝑐 𝑒 𝛾 𝑒 ′ ,𝑡 7 W 𝑐 𝑒 𝑐 𝑒 ′ 𝑡 𝛾 𝑒,𝑒 ′ 𝑡 9 𝑐 𝑢 𝑐 𝑣 𝑐 𝑡 𝛾 𝑢,𝑣 𝑐 𝑡 B b 𝑐 𝑢 𝛾 𝑣,𝑡

			Confluence diagrams
	×1 ×1 𝑢 𝑣 𝑡	𝑐 𝑢 𝛾 𝑣,𝑡	𝛾 𝑢𝑣,𝑡 𝑐 𝑢 𝑐 𝑣𝑡 𝛾 𝑢,𝑣𝑡 𝑐 𝑢𝑣 𝑐 𝑡 A 𝑢,𝑣,𝑡 5 U Õ %	& F 𝑐 𝑢𝑣𝑡 G g
	𝑢 𝑣 𝑡 ×2 ×1
					B 𝑢,𝑣,𝑡 Õ %	𝑐 𝑠 𝑐 𝑠 ′ D d
				6 V 𝑐 𝑢 𝑐 𝑣𝑡	𝛾 𝑢,𝑣𝑡
	𝑢 𝑣 𝑡 ×1 ×2	with 𝑢𝑣 𝑡
			∅	⇝	𝑢𝑣𝑡
					# C
					P p
			𝑢𝑣𝑡 ⇝ ∅

3.3. Theorem. The (3, 1)-polygraph Col 3 (S) is a coherent presentation of the monoid P(S). Proof. Consider the 2-polygraph whose set of 1-cells is Col 1 (S) and whose 2-cells are 𝑐 𝑢 𝑐 𝑣 ⇒ 𝑐 𝑤 𝑐 𝑤 ′ for all 𝑐 𝑢 , 𝑐 𝑣 in Col 1 (S) such that 𝑢 𝑣 × and where 𝑤 and 𝑤 ′ denote respectively the readings of the left and right column of the super tableau (𝑢𝑣 ⇝ ∅) obtained by applying the left insertion. Following Subsection 2.3, by the associativity of the product ★ 𝑟 , the super tableaux (𝑢𝑣 ⇝ ∅) and ⟦𝑢𝑣⟧ 𝑟 are equal, thus this 2-polygraph coincides with Col 2 (S). × 𝑐 𝑢𝑣 𝑐 𝑡 𝛾 𝑢𝑣,𝑡

  1 ,𝑣,𝑡 is constructed as follows. Since 𝑥 𝑝 𝑢 1 ×1 and 𝑢 1 𝑤 × , then X 𝑥 𝑝 ,𝑢 1 ,𝑤 is either of the form A 𝑥 𝑝 ,𝑢 1 ,𝑤 , C 𝑥 𝑝 ,𝑢 1 ,𝑤 or C 𝑥 𝑝 ,𝑢 1 ,𝑤 . Denote 𝑎 1 and 𝑎 ′ 1 the readings of the two columns of ⟦𝑢 1𝑤⟧ 𝑟 . The 3-cell X 𝑥 𝑝 ,𝑢 1 ,𝑤 being confluent, we have 𝑅 𝑐𝑜𝑙 (⟦𝑥 𝑝 𝑎 1 ⟧ 𝑟 ) = 𝑎𝑧 with 𝑧 in S and 𝑅 𝑐𝑜𝑙 (⟦𝑧𝑎 ′ 1 ⟧ 𝑟 ) = 𝑎 ′ . Since 𝑧 𝑎 ′ 𝑣 × ), we deduce that X 𝑧,𝑎 ′ 1 ,𝑤 ′ (resp. X 𝑥 𝑝 ,𝑢 1 ,𝑣 ) is either of the form A 𝑧,𝑎 ′ 1 ,𝑤 ′ , C 𝑧,𝑎 ′ 1 ,𝑤 ′ or C 𝑥 𝑝 ,𝑢 1 ,𝑣 , C 𝑥 𝑝 ,𝑢 1 ,𝑣 or C 𝑥 𝑝 ,𝑢 1 ,𝑣). Denote by 𝑠 and 𝑠 ′ the readings of the two columns of ⟦𝑢 1 𝑣⟧ 𝑟 . The 3-cell X 𝑥 𝑝 ,𝑢 1 ,𝑣 being confluent, we obtain that 𝑅 𝑐𝑜𝑙 (⟦𝑥 𝑝 𝑠⟧ 𝑟 ) = 𝑒𝑦 with 𝑦 in S and 𝑅 𝑐𝑜𝑙 (⟦𝑦𝑠 ′ ⟧ 𝑟 ) = 𝑒 ′ . Since 𝑦 𝑠 ′ ×1 and 𝑠 ′ 𝑡 × , we deduce that X 𝑦,𝑠 ′ ,𝑡 is either of the form A 𝑦,𝑠 ′ ,𝑡 , C 𝑦,𝑠 ′ ,𝑡

	×1	1	and 𝑎 ′ 1 𝑤 ′ ×	(resp. 𝑥 𝑝 𝑢 1 ×1	and 𝑢 1

′ ′ 𝑧,𝑎 ′ 1 ,𝑤 ′ (resp. A ′

  and the 2-cell 𝛾 𝑢,𝑣 is equal to 𝛾 ′ 𝑢,𝑥𝑥 ′ : 𝑐 𝑢 𝑐 𝑥𝑥 ′ ⇒ 𝑐 𝑢𝑥 ′𝑐 𝑥 , showing the claim. Hence, the homotopical reduction 𝑅 Γ 3 reduces Col 3 (S) into a coherent presentation of the monoid P(S) whose set of 2-cells is PreCol 2 (S). Denote by PreCol 3 (S) the extended presentation of the monoid P(S) obtained from PreCol 2 (S) by adjunction of the 3-cells 𝑅 Γ 3 (C 𝑥,𝑣,𝑡 ) for 𝑥 𝑣 𝑡 ×1 ×2 with 𝑥𝑣 𝑡, and 𝑅 Γ 3 (D 𝑥,𝑣,𝑡 ) for 𝑥 𝑣 𝑡 ×2 ×2 . Since the homotopical reduction 𝑅 Γ 3 eliminates the 3-cells of Col 3 (S) of the form A 𝑥,𝑣,𝑡 , B 𝑥,𝑣,𝑡 and C 𝑥,𝑣,𝑡 , we obtain the following result. Super Knuth's coherent presentation. We reduce the coherent presentation PreCol 3 (S) into a coherent presentation of the super plactic monoid P(S) whose underlying 2-polygraph is Knuth 2 (S). The Tietze transformation 𝑇 𝜇←𝛾 ′ : CPC 2 (S) ⊤ → PreCol 2 (S) ⊤ defined in Proposition 2.10 substitutes a 2-cell 𝛾 ′ 𝑥 𝑝 ,𝑥 𝑝-1 ...𝑥 1 for the 2-cell 𝜇 𝑥 𝑝 ...𝑥 1 in 𝐶 2 (S), from the bigger column to the smaller one with respect to the order ≼ deglex . Consider the inverse of this Tietze transformation 𝑇 -1 𝜇←𝛾 ′ that substitutes the 2cell 𝜇 𝑥 𝑝 ...𝑥 1 for the 2-cell 𝛾 ′ 𝑥 𝑝 ,𝑥 𝑝-1 ...𝑥 1 , for each column 𝑥 𝑝 . . . 𝑥 1 such that |𝑥 𝑝 . . . 𝑥 1 | > 2 with respect to ≼ deglex . Denote by CPC 3 (S) the (3, 1)-polygraph whose underlying 2-polygraph is CPC 2 (S) and whose 3-cells are 𝑇 -1 𝜇←𝛾 ′ (𝑅 Γ 3 (C ′ 𝑥,𝑣,𝑡 )) for 𝑥 𝑣 𝑡 ×1 ×2 with 𝑥𝑣 𝑡 and 𝑇 -1 𝜇←𝛾 ′ (𝑅 Γ 3 (D 𝑥,𝑣,𝑡 )) for 𝑥 𝑣 𝑡 ×2 ×2 . In this way, we extend 𝑇 -1 𝜇←𝛾 ′ into a Tietze transformation between PreCol 3 (S) and CPC 3 (S). The (3, 1)polygraph PreCol 3 (S) being a coherent presentation of P(S), we deduce the following result. 3.10. Lemma. The (3, 1)-polygraph CPC 3 (S) is a coherent presentation of the monoid P(S). Tietze transformation 𝑇 𝜂,𝜀←𝛾 ′ : Knuth cc 2 (S) ⊤ → CPC 2 (S) ⊤ defined in the proof of Proposition 2.10 replaces the 2-cells 𝜂 𝑐 𝑥,𝑦,𝑧 and 𝜀 𝑐 𝑥,𝑦,𝑧 in Knuth cc 2 (S) by composite of 2-cells in CPC 2 (S). Consider the inverse of this Tietze transformation 𝑇 -1 𝜂,𝜀←𝛾 ′ : CPC 2 (S) ⊤ -→ Knuth cc 2 (S) ⊤ , that substitutes the 2-cell 𝜂 𝑐 𝑥,𝑦,𝑧 (resp. 𝜀 𝑐 𝑥,𝑦,𝑧 ) for the 2-cell 𝛾 ′ 𝑥,𝑧𝑦 (resp. 𝛾 ′ 𝑦,𝑧𝑥 ). Denote by Knuth cc 3 (S) the (3, 1)-polygraph whose underlying 2-polygraph is Knuth cc 2 (S) and whose set of 3-cells is { 𝑇 -1 𝜂,𝜀←𝛾 ′ (𝑇 -1 𝜇←𝛾 ′ (𝑅 Γ 3 (C 𝜂,𝜀←𝛾 ′ (𝑇 -1 𝜇←𝛾 ′ (𝑅 Γ 3 (D 𝑥,𝑣,𝑡 ))) for 𝑥 𝑣 𝑡 ×2 ×2 }.

	3.9. Super Knuth's coherent presentation
	3.9. The

′ 3.8. Proposition. The (3, 1)-polygraph PreCol 3 (S) is a coherent presentation of the monoid P(S).

′ 𝑥,𝑣,𝑡 ))) for 𝑥 𝑣 𝑡 ×1 ×2 with 𝑥𝑣 𝑡 } ∪ { 𝑇 -1


Estelle Fourat

Tristan Fournier
email: tristan.fournier@ehess.fr

Olivier Lepiller

Reflection: Snatched Commensality: To eat or not to eat together in times of Covid-19 in France 1

During the Covid-19 pandemic, the French Government imposed a strict lockdown from March 17 th to May 11 th 2020. These extraordinary times challenged the social norm of commensality, a practice that is particularly strong and engrained in France. How has lockdown impacted meal-sharing habits? How have the rules and norms of commensality withstood the weakening of social bonds caused by lockdown? Based on a qualitative study of 20 socially diverse households, this essay proposes a typology of adaptive commensal eating patterns: 1) Eating together can provide reassurance by reproducing ordinary routines or by enacting food traditions; 2) Tinkering with lockdown rules leads to creative new forms of commensality, or to an ersatz of commensality; 3) Sharing every meal with the same people can trigger the rejection of commensal practices. These patterns may also represent possible phases in the experience of lockdown, since some individuals shifted from one pattern to another. The final part of the paper discusses the very nature of the social link highlighted through the lens of commensality.

Introduction

From March 17 th to May 11 th 2020 (a total of 56 days), the French Government issued a nationwide stay-at-home order, and closed schools, to prevent the spread of the Covid-19 virus.

People required a certificate (printed off or downloaded onto their smartphone from the official Government website) to leave the house, and could only go out for the following reasons: commuting to and from work (if working from home was not possible); medical appointments; essential family duties such as caring for the elderly; or shopping for "essential" goods including food. For one hour a day, people were also permitted to walk their pet or exercise within a one-kilometer radius.

Overnight, social interactions became fixed and sparse: people living alone had to remain alone, and people living with others (in a family unit or shared household) had to remain together.

How did this disruption of social relations affect commensality -traditionally defined as the practice of sharing food with people in the same location at the same time? Meals constitute a practical tool for building and maintaining social ties [START_REF] Julier | Eating Together: Food, Friendship and Inequality[END_REF]) -especially so in France, where commensality is very marked in daily practices, particularly for lunch and dinner [START_REF] Poulain | The Contemporary Diet in France: 'De-structuration' or From Commensalism to 'Vagabond Feeding[END_REF]. Evidently, as in other European societies [START_REF] Yates | Eating Together and Eating Alone: Meal Arrangements in British Households[END_REF][START_REF] Mestdag | Change and Stability in Commensality Patterns: A Comparative Analysis of Belgian Time-Use Data from 1966, 1999 and 2004[END_REF], several factors indicate -even before the pandemic -a progressive individualization of mealtimes: the time allotted to meals continues to decrease, while both eating out and takeout have become considerably more widespread. However, for French people, eating remains a strong and structuring social norm, a "social affair" [START_REF] Fischler | Commensality, Society and Culture[END_REF])attachment to this norm leads the French to develop a less individualized relation to food than people in other countries (especially the United States), and thus to further devalue the practice of eating alone.

In this essay, we propose to explore how lockdown has affected the rules, and forms, of commensality in France. Thus, we aim to study how lockdown has impacted meal-sharing habits and, in turn, how the rules and norms of commensality have withstood the weakening of social bonds caused by lockdown.

A methodology adapted to lockdown

We conducted a qualitative survey of 20 socially diverse French households, aimed at documenting their food practices over the entire duration of lockdown. Due to social distancing, the survey comprised an online questionnaire and a semi-structured phone interview. The participants were recruited from among our respective extended social networks, and to respect their privacy and anonymity each author only interviewed members of their co-authors' networks. The sample may be too small to draw general conclusions, but it does indicate some interesting trends.

On a practical level, we emailed out two requests for 24-hour recalls of food consumption on days 15 and 30 of lockdown. These featured open and multiple-choice questions on: the circumstances in which food was consumed; involvement in food chores; the quantities of food and drink consumed; and food provisioning. During the final week of lockdown (the end date having been announced by the French Government), we additionally investigated the experience of lockdown before it ended, via a phone interview in which participants discussed: a) the material, social and psychological experience of lockdown, and b) the "food experience" during lockdown (supply, culinary practices, meal organization, commensality).

To compose our sample (n=20), we selected half of the participants from urban locations and the other half from rural settings, assuming that the latter may have a different experience of lockdown due to the proximity of nature and the likelihood of having a garden. We were also careful to select participants with diverse professional statuses: unemployed, student, retired, manager, worker, etc. The sample was composed of 13 women and 7 men, ranging in age from 23 to 90 and living in different types of social configurations: alone (n=4), with a partner (n=4), in a single-parent family (n=3), in a nuclear family (n=5) and with housemates2 (n=4). Ethnicity was not requested, in compliance with French survey protocols, nor was religion.

Three patterns of commensal eating during lockdown

By analyzing these data, we have been able to construct a typology identifying three patterns of commensal practices during lockdown. Each of these is broken down into subpatterns.

1) Reproducing normative meals: Commensality as reassurance

In a very uncertain environment, amid destructured routines and possibly fear of the disease, this pattern embodies the practice of reproducing normative shared meals as reassurance that something remains unchanged.

Mimicking the ordinary. Our first observation was that meals act to punctuate a disrupted daily life. Eating together normalizes the abnormal and helps people find meaning in an unusual situation. A couple of foodies we interviewed live in the countryside, in a stone cottage which they own. He is a market gardener; she is a freelance writer. The couple were expecting a baby, who was born during lockdown. For them, the period was literally a moment of withdrawal and sharing activities around food. On the subject of meals, the man explained: "It was our little daily get-together. We'd cook meals, then sit down to eat, spending an hour together. That was our shared moment. […] For her [his partner's] birthday, I invited her for a meal at our home restaurant." Mealtimes can thus become even more elaborate (i.e. recreating the restaurant experience at home for his partner's birthday), in order to produce a celebratory commensal experience to help soothe the anxiety-provoking experience of lockdown. These two dimensions (normality/festivity) resurface in another interview. A 44-year-old man, a consultant living with his partner in a small apartment in the center of a big city, explained how much meals mattered amid disrupted daily life. His partner, who lost her job at the beginning of lockdown, started to cook new recipes in an effort to add a little escapism to their gloomy routine (before lockdown, they had been frequent restaurant goers, and regularly went out with friends): "Even though it's been a difficult time, especially for her, there was a certain joy in opening the cookbook, choosing a recipe, preparing it, then sampling each new dish together."

In the name of tradition. Leading a normal life was considered so important and necessary that some people upheld tradition by continuing to invite neighbors over on special occasions -as though it were the obvious thing to do, and with no concern for disobeying lockdown rules.

One man (a 50-year-old small business owner) and his family invited neighbors and their children round for Easter, hiding chocolate eggs in the garden for the children and enjoying an aperitif (a pre-dinner drink with nibbles) with the parents: "That was the only time we had people over, it was a wonderful occasion!". This example illustrates how individuals reproduced, almost automatically, a festive tradition involving conviviality around food. In this example, there was no specific hygiene protocol, as the imperative was for life to feel normal.

Thus, the normative nature of the commensal experience led people to suspend, within clearly defined commensal situations, the lockdown rules they were fully accepting otherwise. In so doing, they were acting in accordance with a taken-for-granted form of normality.

2) Tinkering with the rules: New forms of commensality

This pattern concerns the practice of reproducing normative shared meals in different, creative forms to tinker with French public health recommendations. We saw this among particularly self-reflexive people who developed personal ways of adapting the rules. Having been used to eating together, and setting great store by it, they fully intended to continue -even with people who did not belong to their household.

Covid-secure commensality. We observed people having knowingly established protocols to preserve, and sometimes even create, commensal situations. A typical illustration was a family of two adults and two children, living in a comfortable apartment in Paris. Though totally sealed off from the outside world, for the first time they began using their small backyard, where they shared aperitifs -and even meals -with neighbors they had never met before. Lockdown thus became an occasion to expand their social network through food, in a way they deemed safe.

Sometimes, the elaborate hygiene protocol was more rigorous. This was the case, for instance, with one woman (a 30 year-old public service executive), who was living alone in a big city.

As someone used to eating out with friends, she was really struggling to handle the solitude.

So, jumping on the opportunity to share a veg box with a neighbor, the pair quickly decided to prolong the feeling of togetherness over dinner. They set up a comprehensive hygiene protocol to maintain this commensality at a worthwhile cost/benefit ratio: she washed her hands on arrival at his house, dried them with her own towel (which she left there for the duration of lockdown), and took her own cutlery, before sharing food they had each cooked in their respective homes. They accepted the risk because it was, according to them, "worth it": "The loneliness would have done us more harm than the risk of getting the virus. It might not have been smart, but I don't know how we'd have coped otherwise." Remote commensality. Maintaining social ties was facilitated by technology, which made it possible to simulate the experience of eating or drinking together -at the same time and face to face, but in different places -via a screen. One man (a 48-year-old public service employee), living alone in the suburb of a small town, organized very frequent online aperitifs with his friends, having been used to meeting them several times a week in a restaurant. "I have friends I've known for years and years. We meet up, have a drink, and eat together. […] So, every weekend, on Friday, Saturday and Sunday evenings, we have these little video calls together.

It's a nightly ritual". As people were physically apart, there was no need for hygiene protocols, and no risks involved. However, the benefit was only partial because the interactions were, in his words, impoverished compared to "real life". As a result, he and his friends never made it to the next stage: a virtual dinner. "A meal would be too complicated. Some of them have children, so it would require too much organization. It's just about seeing each other, and having a chat and a drink. We'll save the meals for real life!" An ersatz of commensality. Although eating together was not possible, we observed that circulating food items, or pictures of food, enabled people to share food in a way akin to a reallife commensal situation. For one older woman (officially retired, although still working to earn extra money), whose social interactions had decreased over lockdown, a virtual bond with her family became more necessary than ever. Gradually, she began sharing pictures of food with her children, resulting in the creation of a WhatsApp group where they shared recipes and photos of dishes made with seasonal fruits (acacia flower fritters, elderflower cordials, strawberries and cream, etc.). At the start of lockdown, she could barely stomach anything, but she reported having regained her appetite through this process. Another remarkable example is that of a man (a 40-year-old bank employee) living in the heart of one of France's largest cities, who was unable to celebrate Ramadan in his usual way. "Really, you could say that lockdown is an advantage when it comes to observing Ramadan. Usually, I'm constantly tempted by what my colleagues are eating, and I miss the daily café visits with my brother. But with lockdown it's easier, because these temptations no longer exist. However, Ramadan is also a spiritual and celebratory period, and we can't celebrate that side of it any more. We can't go to the mosque, participate in group prayers, or meet in the evening to eat with family and friends."

Nevertheless, he adapted to the unusual situation by creating a remote commensal experience with his mother, in the form of cakes which she baked and had sent to him via his brother.

3) Getting overwhelmed: The flip side of commensality

Being locked down as a couple, with family, or with roommates can lead to daily tensions around food. Lots of people spontaneously reported fantasizing about spending time alone and eating alone. People who spent lockdown with others often experienced a peak in tensions, followed by a rift and new, more individualized eating arrangements. We identified two subpatterns triggered by "forced" commensality.

Social facilitation to overindulge. The first subpattern relates to carnivalesque commensal practices, which could lead to health risks, due to social facilitation -i.e. the fact that being in company can incite people to consume more food and drink, notably alcohol [START_REF] Sobal | Sociability and Meals: Facilitation, Commensality, and Interaction[END_REF][START_REF] Fournier | Suivre ou s'écarter de la prescription diététique. Les effets du 'manger ensemble' et du 'vivre ensemble' chez des personnes hypercholestérolémiques en France[END_REF]). These behaviors of forced commensality were especially pronounced during the first weeks of lockdown, then subsequently decreased. One man (a craftsman in his thirties, living in a house share of 18 people in the countryside), explained that he and his housemates gradually began drinking much more alcohol than usual -ultimately too much, in his view.

"We never had much booze in the house before because it's not included in the collective budget, so we set up a common pot to fund it and started drinking on a regular basis. It was a bit like being on vacation! But I realized we couldn't keep drinking like that. I was drinking every day and that bothered me. And it also becomes expensive".

An overdose of commensality. The second subpattern encompasses attempts to reconfigure social ties which broke down due to excessive contact with others, resulting in an overdose of commensality. A women in her twenties (a project manager for a Parisian company with around thirty employees), who spent lockdown with her boyfriend and four other close friends in a family house in the countryside, recounted how oppressive forced commensality can become over time: "At first, we thought we didn't have to eat together all the time. If anyone wanted to eat on their own, they could. But when it came to it, I didn't feel able to -I think it really excludes you, and it's not ideal for the group". Ultimately, they ended up eating every meal together. She would have liked to eat alone with her boyfriend on occasion, but didn't dare depart from the tacit rule of commensality lest she cause a rift. Conversely, a 39-year-old mother (living in a large house in Toulouse with her partner and their three children), who usually loves to cook the family dinners, became progressively overwhelmed by the tensions provoked by lockdown 3 . Planning and organizing three meals a day for five people -while working from home -took a considerable physical and mental toll, to the point where the dinner table gradually became the place for settling scores. Accusations levied at the children mainly focused on the help they were failing to provide spontaneously (setting the table, clearing the dishes, etc.). After three weeks, they stopped eating breakfast and dinner together. They literally abandoned commensality: "When it came to evening meals, no one had much appetite -at least, not at the same time. At first, I found it hard because I really care about the mealtimes where we all come together, and that was basically no longer the case. It was a case of 'no, I'm not hungry ', or 'no, I'm not coming, I'll come later'. From that point on, I started to say to myself 'come on, just leave it'. So now we put out the serving dishes, they [the children] help themselves, and then do as they please." The burden of being together all the time and eating every meal within the same household unit led people to reject commensality and to seek spaces of freedom.

Conclusion: Commensality and the plasticity of the social link

By studying how commensality was reconfigured during the first French Covid-19 lockdown in the spring of 2020, we have highlighted how people were able to reconstruct the social link under constraints. These constraints (the obligation to stay at home other than for "essential" activities, social distancing requirements, etc.) suppressed or considerably reduced what [START_REF] Sobal | Sociability and Meals: Facilitation, Commensality, and Interaction[END_REF] calls commensal circles (networks of people whom an individual is likely to eat with), while commensal units (the individuals eating together at a given meal) shrunk to the domestic group. This study allowed us to emphasize three main observations, possibly corresponding to three stages of the lockdown commensality experience: 1) the strength of the commensality norm that some people felt almost obliged to perform, despite lockdown constraints, and while generally observing lockdown rules; 2) the creativity of those who tinkered and used "tricks"4 (de [START_REF] Certeau | The Practice of Everyday Life[END_REF], sometimes involving virtual devices, to negotiate these constraints or bend lockdown rules; 3) the narrowing of commensal circles to the domestic commensal unit, which can prove oppressive and may lead the commensal link to break down.

Beyond commensality, this latter observation invites discussion of the very nature of the social link. Indeed, it is often assumed that rituals, such as those at issue in commensality, aim at lubricating the social relations, while ensuring their predictability. In tandem, our study underlines another key aspect of the social link: its plasticity. Not only because the social link is plastic and adaptable, but also because it has to be plastic to keep its elasticity and continue to be effective. This plasticity is nourished by participation in a variety of more or less controlled and predictable commensal situations (involving various commensals, places, times, etc.). But at the time of writing, bars and restaurants are still closed and an overnight curfew after 6 pm is in place. Outside-the-home and public commensal situations are largely outlawed, reducing a discussion space where people can present different views and build common ones.

Commensal eating binds the commensals as much as it keeps away those who are not part of the shared meal [START_REF] Grignon | Commensality and Social Morphology: An Essay of Typology[END_REF]: if this context were to what might the implications be for the social link in the long run? In our opinion, the increasing complexity of commensal rules due to restrictions and the fear of contamination will produce two opposite social behaviours: on the one hand, a withdrawal into oneself resulting in the rarefaction of commensal occasions, and on the other hand, resistance from people resulting in adaptative commensal situations (guests sleeping at home after evening meals, clandestine restaurants, etc.).

Pre-existing flat-sharing, or house shares created for lockdown.

The lockdown increased gender inequalities over home cooking, which are already widely documented under normal circumstances. 25% of French women living in heterosexual couples stated that they were spending more time preparing meals during lockdown (IFOP 2020).

Comprising micro-resistances that constitute spaces of creativity and freedom, as well as possibilities for emancipation.

Acknowledgments

Our thanks to all the respondents, and to Lucy Ackroyd and Cadenza Academic Translations for editing this paper.