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Functorial representation theorems for MV\(_\Delta\) algebras with additional operators. (English) Zbl 0987.06012
Building on his paper “An algebraic approach to propositional fuzzy logic” [J. Logic Lang. Inf. 9, No. 1, 91-124 (2000; Zbl 0942.06006)], in this dense paper the author establishes functorial representations for various varieties of MV-algebras enriched with extra operations. The prototypical representation of this kind was given by the present reviewer in J. Funct. Anal. 65, 15-63 (1986; Zbl 0597.46059), by showing that MV-algebras are categorically equivalent to Abelian lattice-ordered groups with a strong unit. As is well known, the variety of MV-algebras is generated by the unit real interval \([0,1]\) equipped with negation \(1-x\) and truncated sum. As a further natural operation over \([0,1]\) one may consider, e.g., the characteristic function \(\Delta\) of the singleton 1. The resulting variety is denoted \(\text{MV}_\Delta\). By adding the product operation (resp., product and its residuum, which amounts to truncated division) one further obtains \(\text{PMV}_\Delta\) algebras (resp., \(L\Pi\) algebras). The author establishes various categorical equivalences between these varieties and other mathematical structures. For instance, it is proved that the variety of \(L\Pi\) algebras is categorically equivalent to regular commutative \(f\)-rings with unit and a suitable ideal. For background on MV-algebras see the monograph: R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic foundations of many-valued reasoning [Trends in Logic, Studia Logica Library, Vol. 7, Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)].

MSC:
06D35 MV-algebras
03E72 Theory of fuzzy sets, etc.
03B52 Fuzzy logic; logic of vagueness
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