Date of Award

1-1-2018

Document Type

Dissertation

Degree Name

Ph.D.

Organizational Unit

Mathematics

First Advisor

Nikolaos Galatos, Ph.D.

Second Advisor

Michael Kinyon

Third Advisor

Andrew Linshaw

Fourth Advisor

Kimon Valavanis

Keywords

Algebraic logic, Categorical equivalences, Duality theory, Residuated lattices, Substructural logic, Topological dualities

Abstract

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.

Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

Rights Holder

Daniel Wesley Fussner

Provenance

Received from ProQuest

File Format

application/pdf

Language

en

File Size

220 p.

Discipline

Logic, Mathematics

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