Weakly free regular semigroups I. The general case

TL;DR

This paper constructs a universal regular semigroup weakly generated by a set X, proving its existence, defining it via presentation, and solving the word problem, thus advancing the understanding of regular semigroup structures.

Contribution

It introduces a universal regular semigroup weakly generated by X, providing a presentation and solving the word problem, which was previously unaddressed.

Findings

Existence of a universal weakly generated regular semigroup F(X)

F(X) is defined by a presentation

The word problem for F(X) is solved

Abstract

We prove the existence of a regular semigroup F(X) weakly generated by X such that all other regular semigroups weakly generated by X are homomorphic images of F(X). The semigroup F(X) is introduced by a presentation and the word problem for that presentation is solved.