Hulls of ordered algebras: Projectability, strong projectability and lateral completeness

There has been compelling evidence during the past decade that lattice-ordered groups (ℓ-groups) play a far more significant role in the study of algebras of logic than it had been previously anticipated. Their key role has emerged on two fronts: First, a number of research articles have established that some of the most prominent classes of algebras of logic may be viewed as ℓ-groups with a modal operator. Second, and perhaps more importantly, recent research has demonstrated that the foundations of the Conrad Program for ℓ-groups can be profitably extended to a much wider class of algebras, namely the variety of e-cyclic residuated lattices – that is, residuated lattices that satisfy the identity x\e≈e/x. Here, the term Conrad Program refers to Paul Conrad's approach to the study of ℓ-groups that analyzes the structure of individual or classes of ℓ-groups by primarily focusing on their lattices of convex ℓ-subgroups. The present article, building on the aforementioned works, studies existence and uniqueness of the laterally complete, projectable and strongly projectable hulls of e-cyclic residuated lattices. While these hulls first made their appearance in the context of functional analysis, and in particular the theory of Riesz spaces, their introduction into the study of algebras of logic adds new tools and techniques in the area and opens up possibilities for a deep exploration of their logical counterparts.