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

Abstract

A number of research articles have established the significant role of lattice-ordered groups (ℓ-groups) in logic. The fact that underpins these studies is the realization that important algebras of logic may be viewed as ℓ-groups with a modal operator. These connections are just the tip of the iceberg. 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 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 – that is residuated lattices that satisfy the identity x\\e≈e/x. 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.