Substructural logics are systems of reasoning that place more stringent requirements on deduction than classical logic. Whereas classical logic concerns itself with inferences about the truth of propositions, substructural logics describe deductions applying to stricter notions, such as constructability or relevance. Such deductive systems are numerous and diverse, and arise not just in mathematics, but also in computer science, philosophy, and linguistics. Nevertheless, they admit a unified algebraic treatment through a class of lattice-ordered monoids called residuated lattices. These play a role in substructural logics analogous to the role that Boolean algebras play in classical logic.
Here we give a brief survey of substructural logics and residuated lattices, and discuss duality-theoretic techniques for their study. We focus in particular on the case study of Sugihara monoids, a class of idempotent residuated lattices that admits a simple topological duality casting considerable light on its algebraic structure. This topological duality is obtained as a restriction of a natural duality for an appropriately-chosen reduct of Sugihara monoids.