Translational hulls of semigroups of endomorphisms of an algebra

Abstract

We consider the translational hull $\Omega \left(I\right)$ of an arbitrary subsemigroup I of an endomorphism monoid $\mathrm{End}\left(A\right)$ 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 $\mathrm{End}\left(A\right)$, which in the case where I is an ideal is simply $\mathrm{End}\left(A\right)$. 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 $\Omega \left(I\right)$ 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.