Implication in weakly and dually weakly orthomodular lattices

TL;DR

This paper introduces an implication operation in weakly and dually weakly orthomodular lattices, explores their structure, and connects them to residuated structures and generalized measures, enhancing understanding of their algebraic properties.

Contribution

The paper presents a new axiomatization of implication in weakly and dually weakly orthomodular lattices and links these lattices to residuated structures and measures.

Findings

Multiple ways to define implication in these lattices.

Connection established between these lattices and residuated structures.

Characterization of lattices via generalized measures.

Abstract

Weakly orthomodular and dually weakly orthomodular lattices were introduced by the authors in a recent paper. Similarly as for orthomodular lattices we try to introduce an implication in these lattices which can be easily axiomatized and which yields a nice lattice structure. As shown in the paper, this can be realized in several different ways. Moreover, we reveal the connection of weakly and dually weakly orthomodular lattices to residuated structures. Moreover, we provide a characterization of these lattices by means of certain generalized measures.