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Representations and identities of involution Plactic-like monoids arising from the meet of the stalactic congruence and its dual

Bin Bin Han, Wen Ting Zhang, Yan Feng Luo

Abstract

Let $\mathsf{mSt}_n$ be the plactic-like monoid obtained by factoring the free monoid over a finite alphabet $\mathcal{A}_n$ by the meet of the stalactic congruence and its dual. In this paper, we prove that $\mathsf{mSt}_n$ can be equipped with multiple involutions, and divide these involutions into $\lfloor\frac{n}{2}\rfloor+1$ types. A faithful representation of $\mathsf{mSt}_n$ under each of these involutions is obtained. We give transparent combinatorial characterizations of identities for $\mathsf{mSt}_n$ under each involution, and so the finite basis problem and identity checking problem for them are solved.

Representations and identities of involution Plactic-like monoids arising from the meet of the stalactic congruence and its dual

Abstract

Let be the plactic-like monoid obtained by factoring the free monoid over a finite alphabet by the meet of the stalactic congruence and its dual. In this paper, we prove that can be equipped with multiple involutions, and divide these involutions into types. A faithful representation of under each of these involutions is obtained. We give transparent combinatorial characterizations of identities for under each involution, and so the finite basis problem and identity checking problem for them are solved.
Paper Structure (15 sections, 27 theorems, 43 equations, 1 table)

This paper contains 15 sections, 27 theorems, 43 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Words and identities
  4. The left-stalactic and right-stalactic monoids
  5. The meet-stalactic monoid and its involutions
  6. Representations of $\operatorname{\mathsf{mSt}}_n$ with involution
  7. Identities of $(\operatorname{\mathsf{mSt}}_n, ^{\sigma_0})$
  8. Embedding into a direct product of copies of $(\operatorname{\mathsf{mSt}}_2, ^{\sigma_0})$
  9. Characterizations of identities and identity basis
  10. Identities of $(\operatorname{\mathsf{mSt}}_n, ^{\sigma_1})$
  11. Embedding into a direct product of copies of $(\operatorname{\mathsf{mSt}}_3, ^{\sigma_1})$
  12. Characterizations of identities and identity basis
  13. Identities of $(\operatorname{\mathsf{mSt}}_n, ^{\sigma_i})$ for $2\leq i\leq \lfloor\frac{n}{2}\rfloor$
  14. Embedding into a direct product of copies of $(\operatorname{\mathsf{mSt}}_2\times \operatorname{\mathsf{mSt}}_2,^{\rho})$
  15. Characterizations of identities and identity basis

Key Result

Lemma 2.2

Let $\mathbf{u}, \mathbf{v} \in\mathcal{A}_n^{\star}$. Then $\mathbf{u} \equiv_\mathsf{r} \mathbf{v}$ if and only if $\operatorname{\mathsf{ev}}(\mathbf{u})=\operatorname{\mathsf{ev}}(\mathbf{v})$ and $\operatorname{\mathsf{fp}}(\mathbf{u})=\operatorname{\mathsf{fp}}(\mathbf{v})$; and $\mathbf{u} \e

Theorems & Definitions (50)

  • Remark 2.1
  • Lemma 2.2: AR24
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Theorem 3.1: CJKM21
  • Theorem 3.2
  • proof
  • ...and 40 more