125 research outputs found
Categories of Residuated Lattices
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\u27s binary Kripke-style semantics for the relevance logic R-mingle
Poset products as relational models
We introduce a relational semantics based on poset products, and provide
sufficient conditions guaranteeing its soundness and completeness for various
substructural logics. We also demonstrate that our relational semantics unifies
and generalizes two semantics already appearing in the literature: Aguzzoli,
Bianchi, and Marra's temporal flow semantics for H\'ajek's basic logic, and
Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz
logic. As a consequence of our general theory, we recover the soundness and
completeness results of these prior studies in a uniform fashion, and extend
them to infinitely-many other substructural logics
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