Varieties corresponding to classes of complemented posets

Abstract

As algebraic semantics of the logic of quantum mechanics there are usually used orthomodular posets, i.e. bounded posets with a complementation which is an antitone involution and where the join of orthogonal elements exists and the orthomodular law is satisfied. When we omit the condition that the complementation is an antitone involution, then we obtain skew-orthomodular posets. To each such poset we can assign a bounded lambda-lattice in a non-unique way. Bounded lambda-lattices are lattice-like algebras whose operations are not necessarily associative. We prove that any of the following properties for bounded posets with a unary operation can be characterized by certain identities of an arbitrary assigned lambda-lattice: complementarity, orthogonality, almost skew-orthomodularity and skew-orthomodularity. It is shown that these identities are independent. Finally, we show that the variety of skew-orthomodular lambda-lattices is congruence permutable as well as congruence regular.