Translational hulls of semigroups of endomorphisms of an algebra

TL;DR

This paper studies the structure of translational hulls of subsemigroups of endomorphism monoids in universal algebras, establishing conditions for their realizations and connections to idealisers, with applications to various algebraic structures.

Contribution

It provides new conditions for realizing bi-translations as endomorphisms and links translational hulls to idealisers, extending prior work to broader algebraic contexts.

Findings

Conditions for bi-translation realization as endomorphisms

Isomorphisms between translational hulls and idealisers

Methodology for analyzing translational hulls via quotients

Abstract

We consider the translational hull $\Omega(I)$ of an arbitrary subsemigroup $I$ of an endomorphism monoid $\mathrm{End}(A)$ where $A$ is a universal algebra. We give conditions for every bi-translation of $I$ to be realised by transformations, or by endomorphisms, of $A$. We demonstrate that certain of these conditions are also sufficient to provide natural isomorphisms between the translational hull of $I$ and the idealiser of $I$ within $\mathrm{End}(A)$, which in the case where $I$ is an ideal is simply $\mathrm{End}(A)$. We describe the connection between these conditions and work of Petrich and Gluskin in the context of densely embedded ideals. Where the conditions fail, we develop a methodology to extract information concerning $\Omega(I)$ from the translational hull $\Omega(I/{\approx})$ of a quotient $I/{\approx}$ of $I$. We illustrate these concepts in detail in the cases where $A$ is: a free algebra; an independence algebra; a finite symmetric group.