Green's Relations and Dimension in Abstract Semi-groups

This thesis originated in an effort to find an efficient algorithm for the construction of finite inverse semigroups of small order. At one stage in trying to devise such a scheme, an attempt was made to construct an inverse semigroup by adjoining two non-idempotent elements to a semi-lattice in such a way that each of them would be D-equivalent to a pair of distinct D-equivalent idempotents. It was noticed taht such adjunction yielded an inverse semigroup only when the elements of the pari were incomparable in the partial ordering of the semilattice, and only when, for each positive integar n, either both or neither of the elements of the pair had an n-chain of idempotents descending from it. Two theorems on inverse semigroups emerged from this observation; they were subsequently generalized to regular semigroups, and finally to arbitrary semigroups, and in this form they appear herein as Lemma 1.2 and Theorem 1.4.