Abstract

A semigroup is an algebraic structure consisting of a non-empty set together with an associative binary operation. The formal study of semigroups began in the early 20th century. Semigroups are important in many areas of mathematics because they are the abstract algebraic underpinning of “memoryless” systems: time-dependent systems that start from scratch at each iteration. In order to study semihypergroup theory, it is necessary to know about the main concepts of semigroup theory. So, in this chapter, we have a brief excursion into semigroup theory. After some basic definitions and examples, we study regular and inverse semigroups, ideals, bi-ideals, quasi-ideals, homomorphisms, congruence relations, isomorphism theorems, Green’s relations, free semigroups, approximation in semigroups, and ordered semigroups.

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