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.


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


Zbl 1039.06005
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