New-from-old full dualities via axiomatisation

TL;DR

This paper establishes conditions under which different dualities for the same algebra are equivalent, enhancing understanding of natural dualities through universal Horn theory.

Contribution

It provides a main theorem linking alter egos of finite algebras to their duality properties, clarifying when different dualities are essentially the same.

Findings

Main theorem characterizes when two alter egos yield the same duality.

Improves understanding of natural dualities through universal Horn theory.

Applies results to key examples in duality theory.

Abstract

We clarify what it means for two full dualities based on the same algebra to be different. Our main theorem gives conditions on two different alter egos of a finite algebra under which, if one yields a full duality, then the other does too. We use this theorem to obtain a better understanding of several important examples from the theory of natural dualities. Throughout the paper, a fundamental role is played by the universal Horn theory of the dual classes.