An algebraic analysis of implication in non-distributive logics

TL;DR

This paper introduces skew Hilbert algebras, a new algebraic structure that generalizes Hilbert algebras and unifies various logical systems involving implication.

Contribution

It defines skew Hilbert algebras as a natural generalization of Hilbert algebras, linking several important logical structures under a common framework.

Findings

Skew Hilbert algebras unify multiple logical structures with implication.

Basic properties of elements in skew Hilbert algebras are established.

Structural results on skew Hilbert algebras are provided.

Abstract

In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g. (generalized) orthomodular lattices, and MV-algebras, which admit a natural notion of implication. In fact, it turns out that skew Hilbert algebras play a similar role for (strongly) sectionally pseudocomplemented posets as Hilbert algebras do for relatively pseudocomplemented ones. We will discuss basic properties of closed, dense, and weakly dense elements of skew Hilbert algebras, their applications, and we will provide some basic results on their structure theory.