The Conrad Program: From l-groups to algebras of logic

TL;DR

This paper extends the Conrad Program, originally used for lattice-ordered groups, to the setting of e-cyclic residuated lattices, offering new tools for studying algebras of logic through lattice-theoretic properties.

Contribution

It develops a Conrad type approach for algebras of logic, specifically extending lattice-theoretic analysis to e-cyclic residuated lattices, introducing new techniques.

Findings

Extension of Conrad Program to residuated lattices

Enhanced understanding of convex subalgebras in residuated lattices

Introduction of new lattice-theoretic tools for logic algebras

Abstract

A number of research articles have established the significant role of lattice-ordered groups (l-groups) in logic. The purpose of the present article is to lay the groundwork for, and provide significant initial contributions to, the development of a Conrad type approach to the study of algebras of logic. The term Conrad Program refers to Paul Conrad's approach to the study of l-groups, which analyzes the structure of individual l-groups or classes of l-groups by primarily using strictly lattice theoretic properties of their lattices of convex l-subgroups. The present article demonstrates that large parts of the Conrad Program can be profitably extended in the setting of e-cyclic residuated lattices. An indirect benefit of this work is the introduction of new tools and techniques in the study of algebras of logic, and the enhanced role of the lattice of convex subalgebras of a residuated lattice.