Checking Admissibility Using Natural Dualities

Abstract

This paper presents a new method for obtaining small algebras to check the admissibility-equivalently, validity in free algebras-of quasiidentities in a finitely generated quasivariety. Unlike a previous algebraic approach of Metcalfe and Röthlisberger that is feasible only when the relevant free algebra is not too large, this method exploits natural dualities for quasivarieties to work with structures of smaller cardinality and surjective rather than injective morphisms. A number of case studies are described here that could not be be solved using the algebraic approach, including (quasi)varieties of MS-algebras, double Stone algebras, and involutive Stone algebras.

Sugihara algebras and Sugihara monoids: Multisorted dualities

“…Our objective in using duality theory is to be able to study a class of algebras A by setting up a well-behaved dual category equivalence between A and a category X so that problems about A can be faithfully translated into problems about X which one anticipates will be more tractable. Our approach to the admissible rules problem originating in [8] and exploited in [4] relies crucially on the use of strong dualities. As the term is used in natural duality theory [11], a strong duality for a finitely generated quasivariety A = ISP(M) sets up a dual equivalence between A and a category X of topological relational structures, generated as a topological quasivariety by M ∼ , a compatible 'alter ego' for M which is injective in X (the technical details need not concern us here).…”

“…In practice, fullness of a duality is normally obtained at second hand by showing that the duality is strong. In certain applications-and this was crucial in the TSM method for testing admissibility [4,8]-consequences of strongness are required, whereby each of the functors setting up a strong duality converts an embedding to a surjection and a surjection to an embedding. Strongness of a single-sorted duality can be defined in several equivalent ways and the same is to be expected of a multisorted duality.…”

“…Our paper [4] was motivated by a very specific problem: to find a computationally feasible method for studying admissible rules for certain extensions of RM. 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].…”

“…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.…”

Sugihara algebras: Admissibility algebras via the test spaces method

“…A preliminary exploration of the idea of using natural dualities to study admissible rules was undertaken by Cabrer and Metcalfe [5], drawing on well-known natural dualities (for De Morgan algebras, in particular). Encouraged by the evidence in [5], Cabrer et al [6] undertook a more extensive study. The setting is finitely generated quasivarieties ISP(M) for which strong dualities are available.…”