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This paper studies finitely generated quasivarieties of Sugihara algebras.These quasivarieties provide complete algebraic semantics for certainpropositional logics associated with the relevant logic R-mingle. Themotivation for the paper comes from the study of admissible rules. Recentearlier work by the present authors, jointly with Freisberg and Metcalfe, laidthe theoretical foundations for a feasible approach to this problem for a rangeof logics---the Test Spaces Method. The method, based on natural dualitytheory, provides an algorithm to obtain the algebra of minimum size on whichadmissibility of sets of rules can be tested. (In the most general case a setof such algebras may be needed rather than just one.) The method enables us toidentify this `admissibility algebra' for each quasivariety of Sugiharaalgebras which is generated by an algebra whose underlying lattice is a finitechain. To achieve our goals, it was first necessary to develop a (strong)duality for each of these quasivarieties. The dualities promise also to alsoprovide a valuable new tool for studying the structure of Sugihara algebrasmore widely.

Sugihara Algebras: Admissibility Algebras via the Test Spaces Method