On green's fuzzy orthodox semigroups

In the field of research, especially in the logical algebra study of classes of regular semigroup are at most importance. Orthodox semigroups constitutes an important class of regular semigroups. An extension of these concepts was introduced by S. Madhavan on 1978 in his research article "Some results on generalized inverse semigroups” (Madhavan, 1978). A band is a semigroup in which every element is an idempotent. Inverse semigroups constitute the most important and promising classes of semigroups. Such a semigroup is certainly regular. But every regular semigroup need not be an inverse semigroup. If in a regular semigroup, any two idempotents commute, then it is an inverse semigroup. A regular semigroup in which idempotents form a semigroup is called an orthodox semigroup. In this study many results of pre algebra have extended the boarder framework of fuzzy setting. Following the formulation of Green's Fuzzy relations in the work "A study of fuzzy congruence on Green's fuzzy relations" (Hariprakash, 2016) here in this paper, the concept and results of orthodox semigroups are characterized using fuzzy properties. A general theory of logical algebra of fuzzy sets was introduced by Zadeh (Zadeh, 1965). In the paper 'A fuzzy approach to complete Upper Semilattice and complete lower semilattice" discussed the concept of fuzzy congruences relation on a semigroup (Hariprakash, 2016). During the course of this work the study concentrated in special classes of Green’s fuzzy relations. It endeavours to find out a classes (quotient classes) of Green’s Fuzzy Relations as a Fuzzy Orthodox Semigroup and called Green’s Fuzzy Orthodox semigroup. Finally four lammas in Green’s Fuzzy Orthodox semigroup are established.With the help of these lammas the study concludes by finding out a theorem stating a necessary and sufficient condition for quotient classes of regular semigroup to be a Green’s Fuzzy Orthodox semigroup. In particular this work establishes Fuzzy Green's Fuzzy classes is an orthodox semigroup when the semigroup is a fuzzy congruence Green's Fuzzy antisimple orthodox semigroup.