A Category of Ordered Algebras Equivalent to the Category of Multialgebras - Coniglio, Marcelo Esteban; de Toledo, Guilherme Vicentin (Bulletin of the Section of Logic, 2023)
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, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \(Set\) and the category of \(CABA\)s. We modify 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 (which we call a \(bottomless Boolean algebra\)) equipped with a structure of \(Σ\)-algebra compatible with its order (that we call \(ord-algebras\)). Conversely, an ord-algebra over \(Σ\) 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 the category of ord-algebras over \(Σ\). 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.