MV-semirings and their sheaf representations
MV-algebras are the algebraic counterpart of many-valued logic. Recently, Di Nola and Gerla introduced MV-semirings, that is, commutative additively idempotent semirings where every element has a residuum, which is roughly a form of negation. In Theorem 10 of the present paper it is shown that every MV-semiring has a natural structure of a distributive lattice. In every MV-algebra we can define an MV-semiring and conversely, that is, in universal algebra terms, we have a crypto-isomorphism between MV-algebras and MV-semirings, and an isomorphism between the two corresponding categories. An interesting problem, investigated in the paper, is the relation between the ideals of an MV-algebra \(A\) and the ideals of the corresponding MV-semiring \(S\). For instance, the prime spectra of \(A\) and \(S\) are in bijection, but they may have different topologies. Finally, the authors show, building on the notion of a localization of a semiring at a prime ideal, that every MV-algebra \(A\) is the MV-algebra of sections of the Grothendieck sheaf of the semiring reduct of \(A\) (despite the stalks of this sheaf are not MV-semirings, but only commutative additively idempotent semirings).
scientific article; zbMATH DE number 6147380
scientific article