Interior operators in a category: idempotency and heredity
type: Item
instance of: scholarly article
title: Interior operators in a category: idempotency and heredity
zbMATH Open document ID: 1230.18002
published in: Topology and its Applications
publication date: 2011-11-08T00:00:00Z
review text: 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.
reviewed by: Ittay Weiss
Mathematics Subject Classification ID: 18A32
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 18D99
Mathematics Subject Classification ID: 18A05
zbMATH DE Number: 5969483
zbMATH Keywords: categorical topology
zbMATH Keywords: interior operators
zbMATH Keywords: idempotency
zbMATH Keywords: heredity
zbMATH Keywords: weak heredity
author: G. Castellini
MaRDI profile type: Publication
full work available at URL: j.topol.2011.06.030
OpenAlex ID: W2037818704
cites work: Q4693062
cites work: Categorical Closure Operators
cites work: Interior Operators and Topological Connectedness
cites work: Closure operators. I
cites work: Q4859897
cites work: Bases of topological epi-reflections
cites work: Interior Operators in General Categories
DOI: 10.1016/J.TOPOL.2011.06.030
description: scientific article; zbMATH DE number 5969483