Geodesic
A Category of Ordered Algebras Equivalent to the Category of Multialgebras
Bulletin of the Section of Logic, Tome 52 (2023) no. 4, pp. 517-550.

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It is well known that there is a correspondence between sets and complete, atomic Boolean algebras CABAs) 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 CABAs. We modify this result by taking multialgebras over a signature Σ, specifically those whose non-deterministic operations cannot return the empty-set, to CABAs 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.
Keywords: multialgebras, ordered algebras, non-deterministic semantics 
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Coniglio, Marcelo Esteban; de Toledo, Guilherme Vicentin. A Category of Ordered Algebras Equivalent to the Category of Multialgebras. Bulletin of the Section of Logic, Tome 52 (2023) no. 4, pp. 517-550. http://geodesic.mathdoc.fr/item/BSL_2023_52_4_a4/

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