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The Word Problem for $(ω- 1)$-Terms over $\mathrm{DAb}$

Jorge Almeida, Manfred Kufleitner, Jan Philipp Wächter

Abstract

We give a ranker-based description using finite-index congruences for the variety $\boldsymbol{\mathrm{DAb}}$ of finite monoids whose regular $\mathcal{D}$-classes form Abelian groups. This combinatorial description yields a normal form for general pseudowords over $\boldsymbol{\mathrm{DAb}}$. For $(ω- 1)$-terms, this normal form is computable, which yields an algorithm for the word problem for $(ω- 1)$-terms of $\boldsymbol{\mathrm{DAb}}$.

The Word Problem for $(ω- 1)$-Terms over $\mathrm{DAb}$

Abstract

We give a ranker-based description using finite-index congruences for the variety of finite monoids whose regular -classes form Abelian groups. This combinatorial description yields a normal form for general pseudowords over . For -terms, this normal form is computable, which yields an algorithm for the word problem for -terms of .

Paper Structure

This paper contains 8 sections, 20 theorems, 83 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Congruence for $\boldsymbol{\mathrm{DA\!b}}$
  4. A Normal Form for Pseudowords over $\boldsymbol{\mathrm{DA\!b}}$
  5. Definition of the Normal Form
  6. Every Element has a Normal Form
  7. The Normal Form is Unique
  8. The Normal Form of (ω - 1)-Terms can be Computed

Key Result

proposition 2.5

A finite monoid $M$ is in $\boldsymbol{\mathrm{DA\!b}}$ if and only if it satisfies $(xy)^{\omega - 1} = (yx)^{\omega - 1}$. Therefore, any monoid $M \in \boldsymbol{\mathrm{DA\!b}}$ also satisfiesOn a side note, we point out that $(xy)^{\omega} = (yx)^{\omega}$ is the defining relation for $\boldsy

Figures (9)

  • Figure 1: Illustration of \ref{['fct:compatibleIfSmallerAlphabet']}: If we replace $x_2$ by $\tilde{x}_2$ (for $\mathop{\mathrm{alph}}\nolimits \tilde{x}_2 \subseteq \mathop{\mathrm{alph}}\nolimits x_2$), the result will be $\rho$-$\lambda$-compatible to $u$ (for $\rho = X_a X_b X_a$ and $\lambda = Y_a Y_b Y_c Y_b$ in out example).
  • Figure 2: The $\rho$-$\lambda$-split $(w_0, w_1, w_2)$ of $w$ in various cases
  • Figure 3: Illustration of the relation $u \sim_{m, \rho, \lambda} v$. We need to have the same alphabet $\mathop{\mathrm{alph}}\nolimits u_1 = \mathop{\mathrm{alph}}\nolimits v_1$ in the middle and the respective left parts and the respective right parts need to have the same number of $a$s modulo $m$ for all $a \not\in \mathop{\mathrm{alph}}\nolimits(u_1) = \mathop{\mathrm{alph}}\nolimits(v_1)$.
  • Figure 4: The $\rho'$-$\lambda'$-splits of $u$ and $v$.
  • Figure 5: The four cases for the positions of $\rho'$ and $\lambda'$
  • ...and 4 more figures

Theorems & Definitions (66)

  • example 2.1
  • definition 2.2
  • proof
  • proposition 2.5
  • proof
  • lemma 2.6
  • proof
  • lemma 2.7
  • proof
  • proof
  • ...and 56 more