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 ofk×j-rough Heyting algebras. These algebras constitute an extension of Heyting algebras and in the casej= 2 they coincide withk-rough Heyting algebras. In this note, we introduce the notion ofk×j-ideal onk×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 ak×j-rough Heyting algebraA, which is used to construct the localizationk×j-rough Heyting algebrasA𝓕. Furthermore, we prove that thek×j-rough Heyting algebras of fractionsASassociated with a ∧ -closed subsetSofAis ak×j-rough Heyting algebra of localization. Finally, in the finite case we prove thatASis isomorphic to a special subalgebra ofA. Since 3-valued Łukasiewicz –Moisil algebras are a particular case ofk×j-rough Heyting algebras, all these results generalize those obtained in 2005 by Chirtes and Busneag.