, The 2-polygraph ACol 2 (n) is a finite convergent presentation of the symplectic plactic monoid P n (C)

, One remarks that the length of h is smaller than h, then h h. Second case: let h = pc u c v q and h = pc w c w q, with p and q are in ACol 1 (n) * and c u , c v , c w and c w are in ACol 1 (n), where w and w are respectively the readings of the right and left columns of P(uv). h h. Since every application of a 2-cell of ACol 2 (n) yields a -preceding word, addition, we have b = z l

, Since z l x 1 , the tableau P(eb) consists of two columns, that we denote by s and s . Then there is a 2-cell ? e,b : c e c b ? c s c s . In the other hand, we have d = z q . . . z p+1 y p . . . y 1 , d = z p . . . z 1 , s = z l . . . z q+1 y q . . . y p+1 x p . . . x 1 and s = z q . . . z p+1 y p . . . y 1 . Hence a = s, d = s and d = b which yields the confluence the tableau P(uw) consists of two columns, then x 1 z l . In addition, z l is greater than each element of v then y j z l . Hence, in all cases, the tableau P(eb) consists of two columns. On the other hand, using Schensted's algorithm, we called ?-normal if it satisfies ?(w) = w. The normalisation determines a monoid via the defining relation w = ?(w). A normalisation (? 1 , ?) is quadratic if the ?-normality of a 1-cell in ? * 1 only depends on its factors of length two and if we can go from a 1-cell w to the 1-cell ?(w) in finitely many steps, each of which consists in applying ? to some factors of length two. The class of a quadratic normalisation is a pair (x, y) of positive integers which means that one obtains the normal form after at most x steps when starting from the left and y steps from the right. Let ? be the restriction of ? to the set of 1-cells of length two

, Using the notion of quadratic normalisation of monoids, our construction allows us to give a new proof of the termination of the 2-polygraph Col 2 (n) without considering the combinatorial properties of tableaux. Consider the map ? : Col 1 (n) * ? Col 1 (n) * sending a 1-cell in Col 1 (n) * to its unique corresponding tableau, Then (Col, vol.1

.. Knuth's-coherent, . Of, and . Monoids, In this way, we reduce the coherent presentation Col 3 (n) of the monoid P n into the coherent presentation Col 3 (n) of P n , whose underlying 2-polygraph is Col 2 (n) and the 3-cells X u,v,t are those of Col 3 (n), but with (u) = 1. We reduce in 4.2.3 the coherent presentation Col 3 (n) into a coherent presentation PreCol 3 (n) of the plactic monoid P n , whose underlying 2-polygraph is PreCol 2 (n) defined in Chapter 2, Subsection 2.2.2. This reduction is given by a collapsible part defined by a set of 3-cells of Col 3 (n). In a final step, we apply three steps of homotopical reduction on the (3, 1)-polygraph Col 3 (n)

, Let ? be a (3, 1)-polygraph. A collapsible part of ? is a triple ? = (? 2 , ? 3 , ? 4 such that the following conditions are satisfied: i) every ? of every ? k is collapsible, that is, t k?1 well-founded order on the cells of ? such that, for every ? in every ? k

, The homotopical reduction of the (3, 1)-polygraph ? with respect to a collapsible part ? is the Tietze transformation, denoted by R ? , from the (3, 1)-category ? 3 to the (3, 1)-category freely generated by the (3, 1)-polygraph obtained from ? by removing the cells of ? and all the corresponding redundant cells

, ) into a coherent presentation of the monoid P n . The set of 2-cells of this coherent presentation is given by

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