A Development of the Loop Concept

Abstract

The purpose of this thesis is to present in as lucid a manner as possible, the various properties of the abstract binary system defined as a loop. Special emphasis will be placed on the comparative development of other binary systems with the development of loop theory. Whenever possible, analogous theorems and lemmas will be explored in each system, and the necessary changes, both in the proofs and in the theorems themselves, will be particularly noted. The first section of the paper will investigate two generalizations of a loop, groupoids and quasi-groups, and properties will be developed that are particularly relevant to loop theory. In the second section, loops will be discussed in relation to groups; a loop is itself a generalization of a group.