Very true on CBA fuzzy logic

Abstract

ABSTRACT. CBA logic was introduced as a non-associative generalization of the Lukasiewicz many-valued propositional logic. Its algebraic semantic is just the variety of commutative basic algebras. Petr Hájek introduced vt-operators as models for the "very true" connective on fuzzy logics. The aim of the paper is to show possibilities of using vt-operators on commutative basic algebras, especially we show that CBA logic endowed with very true connective is still fuzzy.

Sublattices corresponding to very true operators in commutative basic algebras

“…If a basic algebra A is commutative, we are able to prove a bit more due to the fact that a commutative basic algebra is in fact a (non-associative) residuated lattice. We can repeat this important result from [1,2]: Lemma 2 (Lemma 3.1 in [1]). Let A = (A; ⊕, ¬, 0) be a commutative basic algebra.…”

“…It was proved by M. Botur and the second author that the concept of a very true operator can be extended in these logics and, in a pure axiomatic setting, also in commutative basic algebras. Moreover, both of these concepts are in a one-to-one correspondence, see [1].…”

“…On the other hand, several properties of very true operators on commutative basic algebras have not been revealed in [1] and we feel the opportunity to do it now. In particular, we show that the image of a very true operator in a commutative basic algebra is in fact a sublattice of the induced lattice and, if moreover an idempotency of the operator is assumed, every very true operator is fully determined by this sublattice in the way developed here.…”

Very true operators on MTL-algebras

“…In order to reduce the number of formal concepts, B M elohl K avek and Vyhodil [12] used the so called hedges, which are special cases of very true operators used in reducing the number of formal concepts in concept lattice. Since very true operator was successful in several distinct tasks in various branches of mathematics [10,[12][13][14], it has been extended to other logical algebras such as MV-algebras [15], R`-monoids [16], commutative basic algebras [17], equality algebras [18], effect algebras [19] and so on.…”