We deal with algebras A = (A, ⊕, ¬, 0) of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) A bears a natural lattice structure, (ii) the elements a for which ¬a is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety.

Abstract. The aim of this paper is to characterize BCC-algebras which are term equivalent to MV-algebras. It turns out that they are just the bounded commutative BCCalgebras. Further, we characterize congruence kernels as deductive systems. The explicit description of a principal deductive system enables us to prove that every subdirectly irreducible bounded commutative BCC-algebra is a chain (with respect to the induced order).

We set up an axiomatic system for the logical connective implication within the framework of MV -algebras which generalizes implication in classical logic described similarly by J. C. Abbott. The induced structure (weak implication algebra) turns to be a join-semilattice whose principal filters are MV -algebras.

We present a simple condition under which a bounded lattice L with sectionally antitone involutions becomes an MV-algebra. In this case, L is distributive. However, we get a criterion characterizing distributivity of L in terms of antitone involutions only.

No abstract