A Category of Ordered Algebras Equivalent to the Category of Multialgebras

TL;DR

This paper establishes an equivalence between categories of multialgebras and ordered algebras, extending classical set-Boolean algebra correspondences to non-deterministic algebraic structures with order.

Contribution

It generalizes the set-Boolean algebra correspondence to multialgebras and ordered algebras, introducing a new categorical equivalence for non-deterministic algebraic structures.

Findings

Establishes an equivalence between multialgebras and ordered algebras.

Extends classical Boolean algebra-set correspondence to multialgebras.

Shows non-determinism can be modeled by order structures.

Abstract

It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (CABA's) taking a set to its power-set and, reciprocally, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of $\textbf{Set}$ and the category of CABA's. We extend this result by taking multialgebras over a signature $\Sigma$, specifically those whose non-deterministic operations cannot return the empty-set, to CABA's with their zero element removed and a structure of $\Sigma$-algebra compatible with its order; reciprocally, one of these "almost Boolean" $\Sigma$-algebras is taken to its set of atomic elements equipped with a structure of multialgebra over $\Sigma$. This leads to an equivalence between the category of $\Sigma$-multialgebras and a category of ordered $\Sigma$-algebras. The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.