A Category of Ordered Algebras Equivalent to the Category of Multialgebras

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 $Σ$, specifically those whose non-deterministic operations cannot return the empty-set, to CABA's with their zero element removed and a structure of $Σ$-algebra compatible with its order; reciprocally, one of these "almost Boolean" $Σ$-algebras is taken to its set of atomic elements equipped with a structure of multialgebra over $Σ$. This leads to an equivalence between the category of $Σ$-multialgebras and a category of ordered $Σ$-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.