Effects of semigroup properties on local embeddability

TL;DR

This paper explores how specific properties of semigroups influence their ability to be locally embedded into finite semigroups, providing positive results for some classes and negative for others, and clarifying differences between local embeddability and wrappability.

Contribution

It demonstrates that certain semigroup properties determine local embeddability into finite semigroups, and introduces a new construction distinguishing local embeddability from wrappability.

Findings

Positive embeddability results for completely simple and Clifford semigroups.

Negative embeddability results for -trivial semigroups.

A novel construction showing differences between local embeddability and wrappability.

Abstract

We investigate whether semigroups with a given property which are also locally embeddable into finite semigroups can be locally embedded into finite semigroups with the same property, obtaining a positive answer for completely simple and Clifford semigroups (similarly to group and inverse semigroup cases studied previously) and a negative answer for $\J$-trivial semigroups (similarly to cancellative semigroups). Additionally, we resolve the standing question on the differences between local embeddability in and local wrappability by finite structures, providing a novel construction of $\J$-trivial semigroups which satisfy the latter, but not the former.