Date of Award
Nikolaos Galatos, Ph.D.
Algebraic logic, Categorical equivalences, Duality theory, Residuated lattices, Substructural logic, Topological dualities
We present dual variants of two algebraic constructions of certain classes of residuated lattices: The Galatos-Raftery construction of Sugihara monoids and their bounded expansions, and the Aguzzoli-Flaminio-Ugolini quadruples construction of srDL-algebras. Our dual presentation of these constructions is facilitated by both new algebraic results, and new duality-theoretic tools. On the algebraic front, we provide a complete description of implications among nontrivial distribution properties in the context of lattice-ordered structures equipped with a residuated binary operation. We also offer some new results about forbidden configurations in lattices endowed with an order-reversing involution. On the duality-theoretic front, we present new results on extended Priestley duality in which the ternary relation dualizing a residuated multiplication may be viewed as the graph of a partial function. We also present a new Esakia-like duality for Sugihara monoids in the spirit of Dunn's binary Kripke-style semantics for the relevance logic R-mingle.
Fussner, Daniel Wesley, "Categories of Residuated Lattices" (2018). Electronic Theses and Dissertations. 1527.
Received from ProQuest
Daniel Wesley Fussner