Domination and Closure

TL;DR

This paper explores the relationship between dominating and closure operators, highlighting their properties, interconnections, and applications in social networks and learning spaces, with a focus on categorical and Galois connections.

Contribution

It clarifies the theoretical relationship between dominating and closure operators and identifies conditions under which they form Galois connections, with applications in social networks and learning spaces.

Findings

Every closure operator is dominating.

Dominating operators with categorical pull-backs form Galois connections.

Applications in social networks and learning spaces are proposed.

Abstract

An expansive, monotone operator is dominating; if it is also idempotent it is a closure operator. Although they have distinct properties, these two kinds of discrete operators are also intertwined. Every closure operator is dominating; every dominating operator embodies a closure. Both can be the basis of continuous set transformations. Dominating operators that exhibit categorical pull-back constitute a Galois connection and must be antimatroid closure operators. Applications involving social networks and learning spaces are suggested