Interior and closure operators on bounded commutative residuated \(\ell\)-monoids. (English) Zbl 1227.06014

Rachůnek, Jiří; Švrček, Filip

Discuss. Math., Gen. Algebra Appl. 28, No. 1, 11-27 (2008).

Summary: Topological Boolean algebras are generalizations of topological spaces defined by means of topological closure and interior operators, respectively. In [Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 39, 183–189 (2000; Zbl 1039.06005)] we generalized topological Boolean algebras to closure and interior operators of MV-algebras, which are an algebraic counterpart of the Łukasiewicz infinite-valued logic. In the present paper, these kinds of operators are extended to (and investigated in) the wide class of bounded commutative \(Rl\)-monoids that contains, e.g., the classes of BL-algebras (i.e., algebras of Hájek’s basic fuzzy logic) and Heyting algebras as proper subclasses.

MSC:

06F05	Ordered semigroups and monoids
03G25	Other algebras related to logic
06A15	Galois correspondences, closure operators (in relation to ordered sets)

Citations:

Zbl 1039.06005