Bounded lattices with antitone involutions and properties of MV-algebras.(English)Zbl 1082.03055

The authors consider bounded lattices $$(L;\vee,\wedge, 0,1)$$ where for each element $$p$$ there exists an antitone involution on the interval $$[p,1]$$. They show that such structures are in a one-to-one correspondence with algebras $$(L;\cdot,0)$$ of type $$(2,0)$$ satisfying six identities (similar to that of Abbott’s implication algebras).
Furthermore, they assign to each algebra $$(L;\cdot,0)$$ of that kind an algebra $$(L;\oplus,\neg,0)$$ of type $$(2,1,0)$$ satisfying some of the axioms of MV-algebras. Finally, a condition (the so-called “exchange identity”) for $$(L;\cdot, 0)$$ is given which characterizes the cases in which the assigned algebra $$(L;\oplus,\neg,0)$$ is an MV-algebra.

MSC:

 03G25 Other algebras related to logic 06D35 MV-algebras
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