Embedding in factorisable restriction monoids

TL;DR

This paper proves that restriction semigroups can be embedded into factorisable restriction monoids and introduces a proper cover embeddable in a semidirect product of a semilattice by a group, advancing algebraic structure theory.

Contribution

It establishes the embeddability of restriction semigroups into factorisable restriction monoids and constructs proper covers within semidirect products, providing new insights into their algebraic structure.

Findings

Restriction semigroups are embeddable in factorisable restriction monoids.

Each restriction semigroup has a proper cover embeddable in a semidirect product of a semilattice by a group.

The equivalence between embeddability in factorisable restriction monoids and almost factorisable restriction semigroups.

Abstract

Each restriction semigroup is proved to be embeddable in a factorisable restriction monoid, or, equivalently, in an almost factorisable restriction semigroup. It is also established that each restriction semigroup has a proper cover which is embeddable in a semidirect product of a semilattice by a group.