Relationships among quasivarieties induced by the min networks on inverse semigroups

TL;DR

This paper investigates the complex relationships among various quasivarieties of inverse semigroups generated by specific congruence sequences related to kernels and traces, revealing structural insights into their interconnections.

Contribution

It introduces a sequence of quasivarieties derived from inverse semigroups via congruence kernels and traces, analyzing their interrelations and structural properties.

Findings

Identifies the sequence of quasivarieties generated by congruence kernels and traces.

Establishes relationships and hierarchies among these quasivarieties.

Provides a framework for understanding inverse semigroup congruence structures.

Abstract

A congruence on an inverse semigroup $S$ is determined uniquely by its kernel and trace. Denoting by $\rho_k$ and $\rho_t$ the least congruence on $S$ having the same kernel and the same trace as $\rho$, respectively, and denoting by $\omega$ the universal congruence on $S$, we consider the sequence $\omega$, $\omega_k$, $\omega_t$, $(\omega_k)_t$, $(\omega_t)_k$, $((\omega_k)_t)_k$, $((\omega_t)_k)_t$, $\cdots$. The quotients $\{S/\omega_k\}$, $\{S/\omega_t\}$, $\{S/(\omega_k)_t\}$, $\{S/(\omega_t)_k\}$, $\{S/((\omega_k)_t)_k\}$, $\{S/((\omega_t)_k)_t\}$, $\cdots$, as $S$ runs over all inverse semigroups, form quasivarieties. This article explores the relationships among these quasivarieties.