Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computing congruences of finite inverse semigroups

Luna Elliott, Alex Levine, James D. Mitchell

TL;DR

The paper develops an algorithmic framework for computing congruences on finite inverse semigroups from generating pairs, grounded in the kernel–trace description and Green's relations. It introduces a robust data-structure capturing both the semigroup and its quotients, along with procedures to compute the trace, the $\, ext{H}$-classes of quotients, and individual congruence classes, including efficient membership tests and lattice operations (meets/joins). A specialized method using centralisers computes the maximum idempotent-separating congruence, with practical demonstrations and an implementation showing substantial speedups over prior work. By integrating group theory, automata theory, and inverse semigroup theory, the work provides a scalable route to solving a classical computational problem, albeit not in polynomial time in the worst case.

Abstract

In this paper we present a novel algorithm for computing a congruence on an inverse semigroup from a collection of generating pairs. This algorithm uses a myriad of techniques from the theories of groups, automata, and inverse semigroups. An initial implementation of this algorithm outperforms existing implementations by several orders of magnitude.

Computing congruences of finite inverse semigroups

TL;DR

The paper develops an algorithmic framework for computing congruences on finite inverse semigroups from generating pairs, grounded in the kernel–trace description and Green's relations. It introduces a robust data-structure capturing both the semigroup and its quotients, along with procedures to compute the trace, the -classes of quotients, and individual congruence classes, including efficient membership tests and lattice operations (meets/joins). A specialized method using centralisers computes the maximum idempotent-separating congruence, with practical demonstrations and an implementation showing substantial speedups over prior work. By integrating group theory, automata theory, and inverse semigroup theory, the work provides a scalable route to solving a classical computational problem, albeit not in polynomial time in the worst case.

Abstract

In this paper we present a novel algorithm for computing a congruence on an inverse semigroup from a collection of generating pairs. This algorithm uses a myriad of techniques from the theories of groups, automata, and inverse semigroups. An initial implementation of this algorithm outperforms existing implementations by several orders of magnitude.
Paper Structure (10 sections, 19 theorems, 57 equations, 6 figures)

This paper contains 10 sections, 19 theorems, 57 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A data structure for inverse semigroups and their quotients
  4. Computing the trace
  5. Computing the group $\mathscr{H}$-classes of the quotient
  6. Computing a class of a congruence
  7. Testing membership
  8. Meets and joins
  9. The maximum idempotent-separating congruence
  10. Performance of the algorithms

Key Result

Theorem 2.1

Let $S$ be a finite semigroup and let $a, b\in S$ be such that $(a, b)\in \mathscr{D}$. Then the following are equivalent:

Figures (6)

  • Figure 1: Diagram of the word graph $\Gamma_X$ from \ref{['ex-trace']}, where $x_1$ is represented in magenta, $x_2$ in blue and $x_3$ in orange. Each node is the idempotent that is the identity on the set in its label. Shaded nodes correspond to the idempotents belonging to the only non-singleton class of $\mathop{\mathrm{Tr}}\nolimits(\rho)$.
  • Figure 2: Diagram of the maximum quotient of the word graph $\Gamma_X$ from \ref{['ex-trace']} by the generating pairs of $\mathop{\mathrm{Tr}}\nolimits(\rho)$ from \ref{['lem-gen-pairs-trace']}, where $x_1$ is represented in magenta, $x_2$ in blue and $x_3$ in orange.
  • Figure 3: Comparison of the run-times of an implementation of the algorithm described in \ref{['section-compute-the-trace']} and the earlier implementation in Semigroups described in Torpey for computing the trace of a congruence on an inverse semigroup.
  • Figure 4: Comparison of the run-times of an implementation of the algorithm described in \ref{['section-compute-the-kernel']} and the earlier implementation in Semigroups described in Torpey for computing the kernel and trace of a congruence on an inverse semigroup.
  • Figure 5: Comparison of the run-times of the following algorithms for computing the number of classes of a congruence: the implementation of the algorithm described in \ref{['section-compute-the-trace']}; the earlier implementation in Semigroups described in Torpey using the kernel and trace; and the algorithm implemented in libsemigroups and Semigroups for finding a congruence on a (not necessarily inverse) semigroup.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Theorem 2.1: Location Theorem, cf. Proposition 2.3.7 in Howie
  • Lemma 2.2
  • Lemma 4.1: Generating pairs for the trace
  • proof
  • Lemma 4.2
  • proof
  • Corollary 4.3: Normal congruences as quotients of word graphs
  • Example 4.4
  • Lemma 5.1
  • proof
  • ...and 28 more