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

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)

Zbl 1039.06005
Full Text: