We give a new Esakia-style duality for the category of Sugihara monoids based on the Davey-Werner natural duality for lattices with involution, and use this duality to greatly simplify a construction due to Galatos-Raftery of Sugihara monoids from certain enrichments of their negative cones. Our method of obtaining this simplification is to transport the functors of the Galatos-Raftery construction across our duality, obtaining a vastly more transparent presentation on duals. Because our duality extends Dunn's relational semantics for the logic R-mingle to a categorical equivalence, this also explains the Dunn semantics and its relationship with the more usual Routley-Meyer semantics for relevant logics.
In this paper, we investigate the asymptotic behavior of the Benjamin-Bona-Mahony equation in unbounded domains. We prove the existence of a global attractor when the equation is defined in a three-dimensional channel. The asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions.
In residuated binars there are six non-obvious distributivity identities of ·, /, \ over ∧, ∨. We show that in residuated binars with distributive lattice reducts there are some dependencies among these identities; specifically, there are six pairs of identities that imply another one of these identities, and we provide counterexamples to show that no other dependencies exist among these.Mathematics Subject Classification. 06F05, 03G10, 08B15.
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 Hájek'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.
We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators.
Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces.
In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces.
In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.
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