Implication in sharply paraorthomodular and relatively paraorthomodular posets

TL;DR

This paper explores implication structures in specific classes of paraorthomodular posets, generalizing Greechie's theorems to Kleene lattices and enhancing understanding of their algebraic properties.

Contribution

It introduces a natural notion of implication with adjointness in paraorthomodular posets and generalizes Greechie's amalgamation theorems to Kleene lattices.

Findings

Implication with adjointness is well-defined in these posets.

Generalization of Greechie's theorems to Kleene lattices.

Enhanced algebraic understanding of paraorthomodular structures.

Abstract

In this paper we show that several classes of partially ordered structures having paraorthomodular reducts, or whose sections may be regarded as paraorthomodular posets, admit a quite natural notion of implication, that admits a suitable notion of adjointness. Within this framework, we propose a smooth generalization of celebrated Greechie's theorems on amalgams of finite Boolean algebras to the realm of Kleene lattices.