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

Chajda, Ivan; Emanovský, Peter

Discuss. Math., Gen. Algebra Appl. 24, No. 1, 31-42 (2004).

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.

Reviewer: G. Eigenthaler (Wien)

MSC:

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

Keywords:

bounded lattice; antitone involution; implication algebra; MV-algebra