Categories of Models of R-Mingle
Wesley Fussner, Nick Galatos
TL;DR
This paper introduces a new duality for Sugihara monoids that simplifies their construction and clarifies their semantic relationships within relevant logic frameworks.
Contribution
It presents a novel Esakia-style duality for Sugihara monoids, simplifying their construction and linking Dunn's relational semantics with Routley-Meyer semantics.
Findings
New duality simplifies Sugihara monoid construction
Duality extends Dunn's semantics to a categorical equivalence
Clarifies relationship between different semantic frameworks
Abstract
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.
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