Medina-Droz, Emanuel

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Interior operators in the category of groups
(2015-05) Medina-Droz, Emanuel; Castellini, Gabriele; College of Arts and Sciences - Sciences; Caceres Duque, Luis F.; Ortiz Navarro, Juan A.; Department of Mathematics; Baiges Valentin, Ivan J.
A previously introduced notion of categorical interior operator is studied in the category of groups. The main purpose of this research is to try to find out how many of the general results that hold for a categorical interior operator in topology can be proved in the category of groups, paying particular attention to the notions of connectedness and disconnectedness. Some general properties of interior operators in groups are studied and the notions of discrete, indiscrete, connected and disconnected groups with respect to an interior operator are introduced. The main objective of this work is to discover whether by means of the above notions, a commutative diagram of Galois connections previously presented in the category of topological spaces, can be reconstructed in the group environment. However, unlike the topological case, the lack of commutativity between inverse images and suprema created a big obstacle that, for the time being, could be overcome only by means of two conjectures. Examples are provided.