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. (shrink)

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 Łukasiewicz 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. (shrink)