On the question of embedding a semigroup into an idempotent generated one

TL;DR

This paper introduces a novel embedding method for semigroups into idempotent-generated semigroups of depth 4, using a semidirect product, applicable to regular semigroups and preserving certain properties.

Contribution

It presents a new semidirect product-based embedding of semigroups into semibands of depth 4, extending applicability to non-monoid and regular semigroups.

Findings

Embedding of semigroups into semibands of depth 4

Special embedding for regular semigroups into depth 2

Comparison showing property preservation in embeddings

Abstract

In this paper we present a new embedding of a semigroup into a semiband (idempotent-generated semigroup) of depth 4 (every element is the product of 4 idempotents) using a semidirect product construction. Our embedding does not assume that S is a monoid (although it assumes a weaker condition), and works also for (non-monoid) regular semigroups. In fact, this semidirect product is particularly useful for regular semigroups since we can defined another embedding for these semigroups into a smaller semiband of depth 2. We shall then compare our construction with other known embeddings, and we shall see that some properties of S are preserved by our embedding.