Loomis-Sikorski theorem for \(\sigma\)-complete MV-algebras and \(\ell\)-groups. (English) Zbl 0958.06006

Chang’s MV-algebras were introduced by C. C. Chang in the late fifties as an algebraic counterpart of infinite-valued Łukasiewicz logic. MV-algebras are an important generalization of Boolean algebras, and, as proved by the present reviewer, are also categorically equivalent to abelian lattice-ordered groups with strong unit. (For background 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)].) As a consequence of Chang’s completeness theorem, semisimple MV-algebras are isomorphic to algebras of \([0,1]\)-valued functions. Every MV-algebra \(A\) carries a natural lattice structure \(L(A)\) and when \(L(A)\) is sigma-complete then \(A\) is semisimple. In his paper “Tensor products and the Loomis-Sikorski theorem for MV-algebras” [Adv. Appl. Math. 22, No. 2, 227-248 (1999; Zbl 0926.06004)], the present reviewer established the Loomis-Sikorski theorem for MV-algebras, to the effect that every sigma-complete MV-algebra is the sigma-homomorphic image of some sigma-complete MV-algebra of \([0,1]\)-valued functions. A similar proof is independently given here, together with its variant for Dedekind sigma-complete lattice-ordered abelian groups with strong unit.


06D35 MV-algebras
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces