Ordered Algebras and Logic

Ordered algebras such as Boolean algebras, Heyting algebras, lattice-ordered groups, and MV-algebras have long played a decisive role in logic, although perhaps only in recent years has the significance of the relationship between the two fields begun to be fully recognized and exploited. The first aim of this survey article is to briefly trace the distinct historical roots of ordered algebras and logic, culminating with the theory of algebraizable logics, based on the pioneering work of Lindenbaum and Tarski and Blok and Pigozzi, that demonstrates the complementary nature of the two fields. The second aim is to explain and illustrate the usefulness of this theory, both from an ordered algebra and logic perspective, in the context of the relationship between residuated lattices and substructural logics. In particular, completions on the ordered algebra side, and Gentzen systems on the logic side, are used to address properties such as decidability, interpolation and amalgamation, and completeness.