Natural duality as a tool to study algebras arising from logics (Q2784599)

In this survey paper, the author deals with Davey-Werner dualities for MV-algebras. The latter are an important generalization of Boolean algebras, and were introduced by C. C. Chang in the late fifties as the algebraic counterpart of the infinite-valued Łukasiewicz calculus. For background see the monograph by \textit{R. L. O. Cignoli}, \textit{I. M. L. D'Ottaviano} and \textit{D. Mundici} [Algebraic foundations of many-valued reasoning, Dordrecht: Kluwer (2000; Zbl 0937.06009)]. As the author admits, most results are not new, but his aim is to give simple proofs so as to show the usefulness of such dualities. Unfortunately, bibliographical references are sometimes incomplete. Thus, e.g., Theorem 3 contains a duality result which is similar to Theorem 1 in \textit{R. L. O. Cignoli's} abstract ``Natural dualities for the algebras of Łukasiewicz finite-valued logics'' [Bull. Symb. Log. 2, 218 (1996)] and, as noted by Cignoli himself in his paper, is a particular case of a general result of \textit{D. M. Clark} and \textit{B. A. Davey} [see their monograph ``Natural dualities for the working algebraist'', Cambridge: Cambridge University Press (1998; Zbl 0910.08001), Proposition 3.3.14]. Certain computations in Proposition 1 are due to A. Monteiro [see the above-mentioned book on MV-algebras, page 178]. Problems related to algebraically and existentially closed MV-algebras were also studied by Lacava and others [see, e.g., \textit{F. Lacava} and \textit{D. Saeli} ``Proprieta' e model-completamento di alcune varieta' di algebre di Łukasiewicz'', Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 60, 359-367 (1976; Zbl 0362.02038)]. Related duality results by H. A. Priestley and G. Martinez are ignored [see, e.g., \textit{H. A. Priestley}, ``Natural dualities for varieties of \(n\)-valued Łukasiewicz algebras'', Stud. Log. 54, 333-370 (1995; Zbl 0826.06007); \textit{N. G. Martínez}, ``The Priestley duality for Wajsberg algebras'', Stud. Log. 49, 31-46 (1990; Zbl 0717.03026), ``A simplified duality for implicative lattices of \(\ell\)-groups'', Stud. Log. 56, 185-204 (1996; Zbl 0860.06015)].NEWLINENEWLINEFor the entire collection see [Zbl 0970.00009].