Glivenko like theorems in natural expansions of BCK‐logic

Abstract

Key words Bounded BCK-algebra, involutive BCK-algebra, bounded pocrim, algebraic semantics, natural expansion of a quasivariety, natural expansion of a logic, regular element, Glivenko's theorem, bounded BCKlogic. MSC (2000) 03B47, 03G25, 06F35, 08C15The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity of the analogous of Glivenko theorem in these logics is equivalent to the validity of a simple one-variable formula in the language of BCK-logic with negation.

Interior and closure operators on bounded residuated lattices

“…Recall that the notion of a residuated lattice with GP in the commutative case (as a residuated lattice satisfying the identity ( → ) −− = → −− ) was introduced and investigated in [3].…”

Bounded BCK‐algebras and their generated variety

“…BCK-algebras are defined by (1), (2), (3), and (5). The following properties are well known and can be found in BCK-literature (see for example [2,3,6] and the references given there).…”

“…It follows from the construction given in [3, p. 287] and the results given in [6] that bBCK is just the class of all {→, ⊥, }-subreducts of RL. Moreover, for every R ∈ RL, Con(R) = Con bBCK (R {→, ⊥, }).…”

“…The references given by W. Blok and J. G. Raftery in [2,3] are sufficiently representative. For arithmetical properties on bounded BCK-algebras we use results given in [6].…”

The Localization of Commutative Bounded BCK-Algebras

“…For examples of BCK-algebras see [6][7][8]. If A is a BCK-algebra, then the relation ≤ defined by x y  iff is a partial order on A (which will be called the natural order on A; with respect to this order 1 is the largest element of A.…”