Almost structural completeness; an algebraic approach

Abstract

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.

Singly generated quasivarieties and residuated structures

“…, n} is (is not) unifiable over K. Theorem 7.2 motivates the 'passive' terminology used above, which is adapted from [12]. A complementary demand, now called 'active structural completeness' (ASC) and analysed in [9,18], asks that condition (iii) of Theorem 6.1 should hold for all active quasi-equations; cf. also [64].…”

Deciding active structural completeness

“…[4,32,35]. The reader may consult the monograph [34], the paper [15] and the references therein for many results concerning both properties.…”

“…Hence the replacement of SC by ASC is justified. This problem was undertaken already in [15,30]. It was explicitly formulated in [8], where it was proved that a discriminator variety (which is always ASC) with two distinct constants is SC iff it is minimal or trivial.…”

“…The paper is written in an algebraic language. The notions of admissibility and (A)SC have the algebraic counterparts [1,15]. Thus, they may be studied separately from logic.…”

“…The inclusion ⊇ follows from[15, Corollary 3.2], see also Theorem 7 and the following remark. For the opposite inclusion, observe that G ⊆ H({B × M : B ∈ G}).…”