PARTIALLY ORDERED GAMMA SEMIGROUPS WITH EPIMORPHISMS AND DOMINIONS

Abstract

The purpose of this thesis is to extend some results of partially ordered semigroups to partially ordered gamma semigroups. First important preliminary concepts, examples, lemmas and theorems were presented to make the concept of partially ordered gamma semigroups clear. Many results of semigroups and ordered semigroups are extended to gamma semigroups and ordered gamma semigroups by a number of authors. The concepts of epimorphisms, dominion and zigzag theorem are elucidated by many authors for semigroups and gamma semigroups. In this study the concepts of epimorphisms and dominion for partially ordered semigroup are extended to partially ordered gamma semigroup by using homomorphism between two partially ordered gamma semigroups. Also, Isbell zigzag theorem for partially ordered gamma semigroup have been proved. We showed that subpartially ordered gamma semigroup is epimorphically embedded if partially ordered gamma semigroup identity is preserved underepimorphisms. Also, we proved that dominion of commutative partially ordered gamma semigroup is commutative. By using epimorphism, dominion, zigzag and anti-symmetric property for partially ordered gamma semigroup, we showed that the following permutation identities for partially ordered gamma semigroup: Like 𝑥1γ1𝑥2 = 𝑥2γ1𝑥1, Commutativity (𝐶) 𝑥1γ1𝑥2γ2𝑥3 = 𝑥1γ1𝑥3γ2𝑥2 Right pseudo Commutativity (𝑅𝑃𝐶) 𝑥1γ1𝑥2γ2𝑥3 = 𝑥2γ1𝑥1γ2𝑥3 Left pseudo Commutativity (𝐿𝑃𝐶) 𝑥1γ1𝑥2γ2𝑥3γ3𝑥4 = 𝑥1γ1𝑥3γ2𝑥2γ3𝑥4 Normality (𝑁) are preserved under epimorphisms