On structural completeness versus almost structural completeness problem: A discriminator varieties case study

2019

Arch. Math. Logic

We prove that if an n-element algebra generates the variety V which is actively structurally complete, then the cardinality of the carrier of each subdirectly irreducible algebra in V is at most n (n+1)•n 2•n. As a consequence, with the use of known results, we show that there exist algorithms deciding whether a given finite algebra A generates the (actively) structurally complete variety V(A) in the cases when V(A) is congruence modular or V(A) is congruence meet-semidistributive or A is a semigroup.

Section: Logical Motivationmentioning

confidence: 99%

Deciding active structural completeness

2019

Arch. Math. Logic

“…Varieties of Heyting algebras constitute an adequate semantics for intermediate logics. In particular, the class of all Heyting algebras, which turns out to be a variety, characterizes intuitionistic logic [16,Chapter 7]. As in the case of closure algebras, there is exactly one minimal (quasi)variety of Heyting algebras.…”

Section: Lemma 75 Let V Be a Nontrivial Variety Of Bounded Latticesmentioning

confidence: 99%

“…Open elements of a closure algebra The importance of σ and O class operators follows from the Blok-Esakia theorem. It states that they are mutually inverse lattice isomorphisms between the subvariety lattice of the variety of Heyting algebras and the subvariety lattice of the variety of Grzegorczyk algebras [4, Theorem III.7.10], [16,Theorem 9.66], [85], [29].…”

Section: Asc Without Projective Unification; Modal Companions Of Levimentioning

confidence: 99%

See 1 more Smart Citation

Almost structural completeness; an algebraic approach

Dzik

Annals of Pure and Applied Logic

2016

A deductive system is structurally complete if all of its admissible inference rules are derivable. For several important systems, like the modal logic S5, failure of structural completeness is caused only by the underivability of a passive rule, i.e., a rule whose premise is not unifiable by any substitution. Neglecting passive rules leads to the notion of almost structural completeness, that means, to the derivability of admissible non-passive rules. We investigate almost structural completeness for quasivarieties and varieties of general algebras. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: the one expressed with finitely presented algebras, and the one expressed with subdirectly irreducible algebras. The next one is restricted to quasivarieties with the finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Some connections with exact and projective unification are included.Secondly, examples of almost structurally complete varieties are provided. Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of the modal logic S4. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification is not unitary there. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras.

“…The reader may consult the monograph [60] and references therein for older results concerning SC property. Let us recall here more recent works: [58] about varieties of positive Sugihara monoids, [57] about substructural logics, [17] about fuzzy logics, [70] about some fragment of the intuitionistic logic, [46] about BCK logic, [14] for semisimple varieties and discriminator varieties, and [62] which contains a general considerations from abstract algebraic logic perspective and results for some non-algebraizable deductive systems. Although SC property was often investigated algebraically, there are only few papers about it for algebras not connected to logic.…”

Section: Introductionmentioning

confidence: 99%

Almost structural completeness; an algebraic approach

Dzik

EPiC Series in Computing

Abstract. A deductive system is structurally complete if its admissible inference rules are derivable. For several important systems, like modal logic S5, failure of structural completeness is caused only by the underivability of passive rules, i.e. rules that can not be applied to theorems of the system. Neglecting passive rules leads to the notion of almost structural completeness, that means, derivablity of admissible non-passive rules. Almost structural completeness for quasivarieties and varieties of general algebras is investigated here by purely algebraic means. The results apply to all algebraizable deductive systems.Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: expressed with finitely presented algebras, and with subdirectly irreducible algebras. One is restricted to quasivarieties with finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras.Secondly, examples of almost structurally complete varieties are provided Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of S4 modal logic.A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification in it is not unitary. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras.