Sugihara algebras: Admissibility algebras via the test spaces method

Abstract

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

Sugihara algebras and Sugihara monoids: Multisorted dualities

“…See [27] and [8] and the references therein for background. Compelling evidence for the power of the duality-based approach in [8] was given in [4,Section 8]. There are significant differences between RM and RM t as regards structural completeness properties, with strong assertions being available for the latter.…”

“…There are significant differences between RM and RM t as regards structural completeness properties, with strong assertions being available for the latter. This led us in [4] to focus on the variety SA and specifically on its finitely generated quasivarieties. The generators for these quasivarieties are the algebras Z k for k 1, where Z k has a lattice reduct which is a chain with k+1 2 elements.…”

“…In [4] we presented dualities (specifically natural dualities which have the property of being strong) for all the quasivarieties SA k := ISP(Z k ) of SA . Our representation theorems took different forms in the odd and even cases, with [4, Theorem 6.4] (even case) looking more complicated than [4,Theorem 4.3] (odd case). At the heart of this dichotomy is the fact that the quasivarieties ISP(Z 2m−1 ) are varieties whereas ISP(Z 2m ) HSP(Z 2m ); the variety generated by Z 2m is ISP(Z 2m , Z 2m−1 ).…”

“…We shall assume that the reader is familiar with the basic theory of natural dualities, as summarised in black box fashion in [4,Section 3]. Section 2 below provides a brief formal introduction to the use of multisorted dual categories, with the theory tailored to our needs.…”

“…Once Section 2's theoretical foundations are in place, we focus on Sugihara algebras and monoids. Section 3 recalls facts from [4] on Sugihara algebras and assembles the information we need, likewise, about Sugihara monoids. Section 4 begins with Theorem 4.4, the translation into a two-sorted form of [4,Theorem 4.3].…”