Interior operators in a category: idempotency and heredity
The familiar interior and closure operators in topology are known to be equivalent, and both were generalized to interior and closure operators in a category. In this generality the two approaches are no longer equivalent and, whatever the reason may be, there is considerably more literature on closure operators than there is on interior operators. The paper provides reassuring results that exhibit the elegant mathematics of interior operators. The author recalls what an interior operator is and then considers interior operators that are either idempotent, hereditary, or weakly hereditary. All of these concepts (the latter two are new) are generalizations of familiar properties of the ordinary topological interior operator and are very natural. Then follows an instructive list of 11 interesting examples of interior operators, indicating for each whether it is idempotent or (weakly) hereditary, or whether the question is still open. The rest of the paper is concerned with analyzing the basic lattice theoretic properties, in the lattice of all interior operators, of the three sublattices of the idempotent, the hereditary, and the weakly hereditary interior operators. In particular it is shown that the idempotent interior operators are coreflective with a coreflector given explicitly by the idempotent core construction. The hereditary (resp. weakly hereditary) interior operators are reflective with a reflector given explicitly by the hereditary (resp. weakly hereditary) hull construction.
scientific article; zbMATH DE number 5969483