2022/09/11 by Coniglio, Marcelo E., Toledo, Guilherme V.
#Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2209.08158
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 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.