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In this article we have constructed some examples of some classes of AG-groupoids

In this paper, we have investigated different classes of an AG-groupoid by their structural properties. We have prove that weakly regular, intra-regular, right regular, left regular, left quasi regular and completely regular coincide in an AG-groupoid with left identity and in AG^{**}-groupoid. Further we have prove that every regular (weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular) AG-groupoid with left identity (AG^{**}-groupoid) is regular but the converse is not true in general. Also it has been shown that non-associative regular, weakly regular, intra-regular, right regular, left regular, left quasi regular and completely regular AG^{*} groupoids do not exist.

In this paper, we have discussed the properties of intuitionistic fuzzy ideals of an AG-groupoids. We have characterized an intra-regular AG-groupoid in terms of intuitionistic fuzzy left (right, two-sided) ideals, fuzzy (generalized) bi-ideals, intuitionistic fuzzy interior ideals and intuitionistic fuzzy quasi ideals. We have proved that the intuitionistic fuzzy left (right, interior, quasi) ideal coincides in an intra-regular AG-groupoid. We have also shown that the set of intuitionistic fuzzy two-sided ideals of an intra-regular AG-groupoid forms a semilattice structure.

In this paper we have discusses ��-left, ��-right, ��-bi-, ��-quasi-, ��-interior and ��-ideals in ��-AG^{**}-groupoids and regular ��-AG^{**}-groupoids. Moreover we have proved that the set of ��-ideals in a regular ��-AG^{**}-groupoid form a semilattice structure. Also we have characterized a regular ��-AG^{**}-groupoid in terms of left ideals., This paper has been withdrawn by the author due to some problems

In this paper, we have introduced the concept of $\left( \in ,\in \vee q\right) $-fuzzy ideals in a right modular groupoid. We have discussed several important features of a completely regular right modular groupoid by using the $\left( \in ,\in \vee q\right) $-fuzzy left (right, two-sided) ideals, $\left( \in ,\in \vee q\right) $-fuzzy (generalized) bi-ideals and $\left( \in ,\in \vee q\right) $-fuzzy $(1,2)$-ideals. We have also used the concept of $\left( \in ,\in \vee q_{k}\right) $-fuzzy left (right, two-sided) ideals, $\left( \in ,\in \vee q_{k}\right) $-fuzzy quasi-ideals $\left( \in ,\in \vee q_{k}\right) $-fuzzy bi-ideals and $% \left( \in ,\in \vee q_{k}\right) $-fuzzy interior ideals in completely regular right modular groupoid and proved that the $\left( \in ,\in \vee q_{k}\right) $-fuzzy left (right, two-sided), $\left( \in ,\in \vee q_{k}\right) $-fuzzy (generalized) bi-ideals, and $\left( \in ,\in \vee q_{k}\right) $-fuzzy interior ideals coincide in a completely regular right modular groupoid.

Using the idea of a quasi-coincedece of a fuzzy point with a fuzzy set, the concept of an ({\alpha},{\beta})-fuzzy bi-ibeals in AG-groupoid is introduced in this paper, which is a generalization of the concept of a fuzzy bi-ideal in AG-groupoid and some interesting characterizations theorems are obtained.

We have characterized an intra-regular {\Gamma}-AG^{**}-groupoids by using the properties of {\Gamma}-ideals (left, right, two-sided ), {\Gamma}-interior, {\Gamma}-quasi, {\Gamma}-bi and {\Gamma}-generalized bi and {\Gamma}-(1,2)). We have prove that all the {\Gamma}-ideals coincides in an intra-regular {\Gamma}-AG^{**}-groupoids. It has been examined that all the {\Gamma}-ideals of an intra-regular {\Gamma}-AG^{**}-groupoids are {\Gamma}-idempotent. In this paper we define all {\Gamma}-ideals in {\Gamma}-AG^{**}-groupoids and we generalize some results.

In this paper, we have introduced the notion of ��-fuzzification in ��-AG-groupoids which is in fact the generalization of fuzzy AG-groupoids. We have studied several properties of an intra-regular ��-AG^{**}-groupoids in terms of fuzzy ��-left (right, two-sided, quasi, interior, generalized bi-, bi-) ideals. We have proved that all fuzzy ��-ideals coincide in intra-regular ��-AG^{**}-groupoids. We have also shown that the set of fuzzy ��-two-sided ideals of an intra-regular ��-AG^{**}-groupoid forms a semilattice structure.

In this paper we have discusses {\Gamma}-left, {\Gamma}-right, {\Gamma}-bi-, {\Gamma}-quasi-, {\Gamma}-interior and {\Gamma}-ideals in {\Gamma}-AG^{**}-groupoids and regular {\Gamma}-AG^{**}-groupoids. Moreover we have proved that the set of {\Gamma}-ideals in a regular {\Gamma}-AG^{**}-groupoid form a semilattice structure. Also we have characterized a regular {\Gamma}-AG^{**}-groupoid in terms of left ideals.

In this paper, we have introduced the notion of (1,2)-ideal in an LA-semigroup and shown that (1,2)-ideal and two-sided ideal coincide in an intra-regular LA-semigroup. We have characterized an intra-regular LA-semigroup by using the properties of left and right ideals. Some natural examples of LA-semigroups have been given. Further we have investigated some useful conditions for an LA-semigroup to become an intra-regular LA-semigroup and given the counter examples to illustrate the converse inclusions. All the ideals (left, right, two-sided, interior, quasi, bi- generalized bi- and (1,2)) of an intra-regular LA-semigroup have been characterized. Finally we have given an equivalent statement for a two-sided ideal of an intra-regular LA-semigroup in terms of the intersection of two minimal two-sided ideals of an intra-regular LA-semigroup.

In this paper, we have introduced the concept of intuitionistic fuzzy ideals in an AG-groupoids. We have characterized regular and intra-regular AG-groupoids in terms of intuitionistic fuzzy left (right, two-sided) ideals, fuzzy (generalized) bi-ideals and intuitionistic fuzzy (1,2)-ideals. We have proved that all the intuitionistic fuzzy ideals coincides in regular and intra-regular AG-groupoids. It has been shown that the set of intuitionistic fuzzy two sided ideals of a regular AG-groupoid forms a semilattice structure. We have also given some useful conditions for an AG-groupoid to become an intra-regular AG-groupoid in terms their intuitionistic fuzzy ideals.

In this paper, we have discussed several properties of intuitionistic fuzzy ��-ideals of a ��-AG-groupoid which is the generalization of ideals in AG-groupoid We have characterized an intra-regular ��-AG^{**}-groupoid in terms of intuitionistic fuzzy ��-left (right, two-sided) ideals, intuitionistic fuzzy (��-generalized bi) ��-bi-ideals, intuitionistic fuzzy ��-interior ideals and intuitionistic fuzzy ��-quasi ideals. We have proved that the intuitionistic fuzzy ��-left (right, interior, quasi) ideals coincide in an intra-regular ��-AG^{**}-groupoid. We have also shown that the set of intuitionistic fuzzy ��-two-sided ideals of an intra-regular ��-AG^{**}-groupoid forms a semilattice structure.

A left almost semigroup (LA-semigroup) or an Abel-Grassmann's groupoid (AG-groupoid) is investigated in several papers. In this paper we have discussed ideals in LA-semigroups. Specifically, we have shown that every ideal in an LA-semigroup S with left identity e is prime if and only if it is idempotent and the set of ideals of S is totally ordered under inclusion. We have shown that an ideal of S is prime if and only if it is semiprime and strongly irreducible. We have proved also that every ideal in a regular LA-semigroup S is prime if and only if the set of ideals of S is totally ordered under inclusion. We have proved in the end that every ideal in S is prime if and only if it is strongly irreducible and the set of ideals of S form a semilattice., 6 pages, 1table

In this paper we have discussed the ideals in Abel Grassmann's groupoids and construct their topologies., 5pages

In the present paper we have studied the concept of fuzzification in AG-groupoids. The equivalent statement for an AG-groupoid to be a commutative semigroup is proved. Fuzzy points have been defined in an AG-groupoid and has been shown the representation of smallest fuzzy left ideal generated by a fuzzy point. The set of all fuzzy left ideals, which are idempotents, forms a commutative monoid. The relation of fuzzy left(right) ideals, fuzzy interior ideals and fuzzy bi-ideals in AG-groupoid has been studied. Necessary and sufficient condition of fully fuzzy prime AG-groupoid has been shown. Further, It has been shown that the set of fuzzy quasi-prime ideals of AG-groupoid with left identity forms a semillattice structure. Moreover, equivalent statements for fuzzy semiprime left ideal in an AG-groupoid have been proved., 9 pages