Left residuated operators induced by posets with a unary operation

TL;DR

This paper investigates various quantum-inspired posets with unary operations, demonstrating their operator left residuation properties and their potential as algebraic semantics for quantum logic and connections to substructural logics.

Contribution

It introduces conditions under which modified quantum structures are operator left residuated, expanding the understanding of their algebraic properties and logical applications.

Findings

Orthomodular, pseudo-orthomodular, pseudo-Boolean, and Boolean posets are shown to be operator left residuated.

Conditions for operator residuation are both sufficient and necessary when considering subsets.

These posets can serve as algebraic semantics for quantum mechanics and relate to substructural logics.

Abstract

The concept of operator left residuation has been introduced by the authors in a previous paper. Modifications of so-called quantum structures, in particular orthomodular posets, like pseudo-orthomodular, pseudo-Boolean and Boolean posets are investigated here in order to show that they are operator left residuated or even operator residuated. In fact they satisfy more general sufficient conditions for operator residuation assumed for bounded posets equipped with a unary operation. It is shown that these conditions may be also necessary if a generalized version using subsets instead of single elements is considered. The above listed posets can serve as an algebraic semantics for the logic of quantum mechanics in a broad sense. Moreover, our approach shows connections to substructural logics via the considered residuation.