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.


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