Logical and algebraic properties of generalized orthomodular posets

TL;DR

This paper explores the logical and algebraic structures of generalized orthomodular posets, their representations, and conditions for their conversion into other algebraic structures, advancing the mathematical foundations of quantum logic.

Contribution

It introduces new algebraic and logical properties of generalized orthomodular posets, including their representation via algebras and conditions for conversion into operator residuated structures.

Findings

Conditions for conversion into operator residuated structures

Representation of posets by algebras (directoids)

Dedekind-MacNeille completion results

Abstract

Generalized orthomodular posets were introduced recently by D. Fazio, A. Ledda and the first author of the present paper in order to establish a useful tool for studying the logic of quantum mechanics. They investigated structural properties of these posets. In the present paper we study logical and algebraic properties of these posets. In particular, we investigate conditions under which they can be converted into operator residuated structures. Further, we study their representation by means of algebras (directoids) with everywhere defined operations. We prove congruence properties for the class of algebras assigned to generalized orthomodular posets and, in particular, for a subvariety of this class determined by a simple identity. Finally, in contrast to the fact that the Dedekind-MacNeille completion of an orthomodular poset need not be an orthomodular lattice we show that the Dedekind-MacNeille completion of a stronger version of a generalized orthomodular poset is nearly an orthomodular lattice.