Axiomatizing groups in terms of semigroups and quasigroups

Abstract

This research has been done to axiomatize groups in terms of semigroups and quasigroups. In Chapters I through IV, the properties that may be placed on semigroups have been investigated. Chapter I places properties such as one-sided identities, one-sided inverses, unique idempotent, cancellation and divisibility on a semigroup to make it a group. Chapter II discusses the periodic and finite semigroups and relates these two groups. The important concept of ideals is discussed in Chapter III. It is shown that a group has no proper left, right, or two-sided ideals. Chapter IV investigates the fact that Scan have no proper bi-ideals and quasi-ideals and examples are constructed to show their behavior in semigroups and groups. Lastly, Chapter V axiomatizes groups in terms of quasigroups. This amounts to investigating the important property of associativity and using a generalized associative law to get a group.