Varieties corresponding to classes of complemented posets

TL;DR

This paper explores algebraic structures called skew-orthomodular posets, their associated lambda-lattices, and characterizes properties like complementarity and orthomodularity through identities, establishing their algebraic independence and congruence properties.

Contribution

It introduces a new class of algebraic structures called skew-orthomodular lambda-lattices and characterizes key properties via identities, expanding the algebraic understanding of quantum logic semantics.

Findings

Identifies identities characterizing complementarity and orthogonality.

Shows independence of these identities.

Proves the variety of skew-orthomodular lambda-lattices is congruence permutable and regular.

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.