Abstract
This article presents some properties of a special class of interior operators generated by ideals. The mathematical framework is given by complete domains, namely complete posets in which the set of minimal elements is a basis. The first part of the paper presents some preliminary results; in the second part we present the novel interior operator denoted by G(i,I), an operator built starting from an interior operator i and an ideal I. Various properties of this operator are presented; in particular, the connection between the properties of the ideal I and the properties of the operator G(i,I). Two such properties (denoted by Pi and Qi) are extensively analyzed and characterized. Additionally, some characterizations for compact elements are presented.
Highlights
In potential theory and in the study of harmonic functions, the fine topology is the natural topology for subharmonic functions [1]
A good example of a fine topology is given by the density topology on the real line
Let v be a partial order on min( X ) such that (min( X ), v) is an algebraic domain, τ be the density topology, α ≥ 2 be a cardinal number and I be the ideal of all α-unions of τ-nowhere dense subsets of min( X )
Summary
In potential theory and in the study of harmonic functions, the fine topology is the natural topology for subharmonic functions [1]. The density topology is a good example of a topology generated by ideals. The ideals used to generate the density topology is given by the negligible Lebesgue sets; more details are presented in [1]. The advantage of the meagre sets is that the whole construction could be used in any topological space, and does not depend on the properties of the measure. After these approaches, new ideals were considered to generate specific topologies. Inspired by the developments in potential and topology theory, we investigate the notion of interior generated by an ideal in the framework of domain theory. This article presents the results obtained using this novel approach
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