Constrained abstract representation problems in semigroups and partial groupoids

Abstract

In this paper dieren t constrained abstract representa- tion theorems for partial groupoids and semigroups are proved. Methods for improving the retract properties of the structures are also developed. These have strong class-theoretical implications for many types of gener- alized periodic semigroups and related partial semigroups. The results are signican t in a model-theoretical setting. In this paper dieren t new methods for improving the retract properties of semigroups and partial semigroups are developed. The results are of much signicance in model-theoretical settings too. Given an ideal or a generalised ideal in a semigroup and two or more subsemigroups subject to certain restric- tions, new semigroups are derived over the same base set via multiple derived partial semigroups. The method is signican t from the classication and em- beddability viewpoints too. The connections between the retract properties of the structures is then shown to be signican t. The results are extended to CSM-partial semigroups -these being dened by the constraints on their process of generation (12). Within the classes of semigroups/partial semigroups for which the method is denable a classication theory appears possible. This directly relates to the automorphism group of the structures. New class-closure operators are introduced in the light of the above. From the model-theoretic viewpoint