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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.

##### MSC:
 06D35 MV-algebras 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
##### Citations:
Zbl 0937.06009; Zbl 0926.06004