Admissibility in Finitely Generated Quasivarieties

Abstract

Abstract. Checking the admissibility of quasiequations in a finitely generated (i.e., generated by a finite set of finite algebras) quasivariety Q amounts to checking validity in a suitable finite free algebra of the quasivariety, and is therefore decidable. However, since free algebras may be large even for small sets of small algebras and very few generators, this naive method for checking admissibility in Q is not computationally feasible. In this paper, algorithms are introduced that generate a minimal (with respect to a multiset well-ordering on their cardinalities) finite set of algebras such that the validity of a quasiequation in this set corresponds to admissibility of the quasiequation in Q. In particular, structural completeness (validity and admissibility coincide) and almost structural completeness (validity and admissibility coincide for quasiequations with unifiable premises) can be checked. The algorithms are illustrated with a selection of well-known finitely generated quasivarieties, and adapted to handle also admissibility of rules in finite-valued logics.

Sugihara algebras: Admissibility algebras via the test spaces method

“…It is easy to see that p s is surjective, so that Z k ∈ H(B k ). From this we can prove algebraically that ISP(B k ) = ISP(F SA k (s)) (see [22,Corollary 22]). This direct, duality-free, argument confirms that B k can be employed for admissibility testing, but does not yield the stronger result that is the minimal such algebra.…”

“…In [24, Section 4.5] SA 3 was proved to be almost structurally complete. The proof that SA 4 is almost structurally complete follows from [22,Theorem 18] combined with the fact that its admissibility algebra is B 4 = Z 4 × Z 2 ).…”

“…When this set is a singleton, we refer to its unique element as the admissibility algebra. In [22] and [24] an algorithm was developed to calculate the admissibility set. This algorithm is implemented in the TAFA package 1 .…”

Varieties of Positive Modal Algebras and Structural Completeness

“…In [4,20,43,59] the following characterizations of the various kinds of structural completeness are obtained. Theorem 1.1.…”

Checking Admissibility Using Natural Dualities

“…This issue of feasibility is addressed by Metcalfe and Röthlisberger in [28]. These authors provide algorithms that for a finite set K of n-generated algebras generating a quasivariety Q, produce a finite set of "small" algebras that admits the same valid quasi-identities as F Q (n), that is, the admissible quasi-identities of Q.…”