A Study on the Relationship Between Minimal Generating Sets and Independence in Semigroups

Abstract

This paperreviews the work ofM.I. Sampson et. Al (2023) [7], and delves into the intricate relationship between minimal generating sets and independence in semigroups by examining the comparability of elements induced by orderings on the semigroup. It demonstrates that the existence of a minimal generating set implies independence, and conversely, independence implies the existence of a minimal generating set. Additionally, the paper presents two new algorithms: one for determining minimal generating sets for countable systems of semigroups and another for any given semigroup. These algorithms offer practical solutions for semigroup theorists and researchers. Through rigorous mathematical analysis and proof, this paper sheds light on the fundamental properties of semigroups and their generating sets.

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