Localization ofk×j-rough Heyting algebras

Abstract

k-rough Heyting algebras were introduced by Eric San Juan in 2008 as an algebraic formalism for reasoning on finite increasing sequences over Boolean algebras in general and on generalizations of rough set concepts in particular. In 2020, we defined and studied the variety of k × j-rough Heyting algebras. These algebras constitute an extension of Heyting algebras and in the case j = 2 they coincide with k-rough Heyting algebras. In this note, we introduce the notion of k × j-ideal on k × j-rough Heyting algebras which allows us to consider a topology of them. Besides, we define the concept of 𝓕-multiplier, where 𝓕 is a topology on a k × j-rough Heyting algebra A, which is used to construct the localization k × j-rough Heyting algebras A 𝓕. Furthermore, we prove that the k × j-rough Heyting algebras of fractions A S associated with a ∧ -closed subset S of A is a k × j-rough Heyting algebra of localization. Finally, in the finite case we prove that A S is isomorphic to a special subalgebra of A. Since 3-valued Łukasiewicz –Moisil algebras are a particular case of k × j-rough Heyting algebras, all these results generalize those obtained in 2005 by Chirtes and Busneag.