Groupoids and semigroupoids

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] The theory of semigroupoids and groupoids makes the transition between arbitrary sets and groups. The usefulness of developing theory stems from various applications where a group structure is lacking, yet there exists some "milder" structure on the set considered. There are numerous applications of semigroupoids/groupoids in: metrization theory, functional analysis, topology, and physics, among others. The material in this thesis is the result of the author's getting acquainted with developments in the theory of semigroupoids/groupoids while reading the recent monograph "Groupoid Metrization Theory With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis" by D. Mitrea, I. Mitrea, M. Mitrea, and S. Monniaux, various parts of which are reproduced in this thesis. In the process, several new avenues are explored for the first time, which contribute to a fuller understanding of this dynamic field.