Endomorphisms and Translations of Semigroups

This thesis is a study of some of the properties of mappings or semigroups into semigroups. In Section 1, definitions and basic concepts are given, with brief mention of some well known properties of semigroups and of mappings. Section 2 is devoted to a few theroems which are useful in the process of finding the semigroup of all endomorphisms of a given semigroup. When theorems on endomorphisms are valid in the more general situation in which a semigroup is mapped homomorphically on another semigroup, we state and prove the theorems in the more general setting. Section 3 consists of some theroems and a short discussion of right and of left translations of semigroups. The rest of the paper is devoted to listing, in section 4, 1) the semigroups of all endomorphisms of semigroups of orders two, three and four, and 2) the semigroups of all right translations of and all left translations on semigroups of orders two, three and four.