Interior operators in general categories

Abstract

Since the introduction of closure operators in topology (Riesz [1909], Kuratowski [1922] and others) many authors have defined and studied closure operators in other branches of mathematics (e.g. Birkhoff [1937] in Algebra and Birkhoff [1940] in Lattice Theory) and in more general contexts. In particular, the introduction of the categorical notion of closure operators (e.g. see Dikranjan and Giuli [1987]) has unified various important notions and has led to interesting examples and applications in diverse areas of mathematics (see for example, Dikranjan and Tholen [1995]). For a topological space it is well–known that the associated closure and interior operators provide equivalent descriptions of the topology, but this is not true in general. So, it makes sense to define and study the notion of interior operators in a general context for its own sake. In a previous paper (Vorster [1997]) we concentrated on the notion of general interior operators in lattices, while in this paper we define and study a categorical notion of interior operators. We also introduce the idea of categorical transformation operators which establishes a relationship between categorical interior and closure operators. Furthermore, we provide some examples and derive some basic properties of categorical interior operators.

Keywords: closure operator; interior operator; transformation operator

Quaestiones Mathematicae 23(2000), 405–416