Regular semigroups weakly generated by idempotents

TL;DR

This paper constructs a universal regular semigroup weakly generated by a set of idempotents, demonstrating its structure, decidability of the word problem, and its role as a dividing semigroup for all finite semigroups.

Contribution

It introduces the semigroup FI(X) as a universal object for regular semigroups weakly generated by idempotents, with a detailed structural analysis and decidability results.

Findings

FI(X) is a universal regular semigroup weakly generated by |X| idempotents.

The word problem for FI(X) is decidable despite infinite generating sets.

FI_2 contains all FI_n as subsemigroups, linking finite and infinite structures.

Abstract

A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation $\langle G(X),\rho_e\cup\rho_s\rangle$ and its structure is studied. Although each of the sets $G(X)$, $\rho_e$, and $\rho_s$ is infinite for $|X|\geq 2$, we show that the word problem is decidable as each congruence class has a canonical form. If $FI_n$ denotes FI(X) for $|X|=n$, we prove also that $FI_2$ contains copies of all $FI_n$ as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide $FI_2$; and (ii) all finite semigroups divide $FI_2$.