Intra-regular Abel-Grassmann's groupoids
We characterize intra-regular Abel-Grassmann's groupoids by the properties of their ideals and $(\in ,\in!\vee q_{k})$-fuzzy ideals of various types.
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arXiv subjects
Madad Khan
Publications and source records attributed to Madad Khan.
Examples of Abel Grassmann's groupoids
In this article we have constructed some examples of some classes of AG-groupoids
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Ideals in intra-regular left almost semigroups
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.
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Characterizations of Γ-AG^{**}-groupoids by their Γ-ideals
In this paper we have discusses Γ-left, Γ-right, Γ-bi-, Γ-quasi-, Γ-interior and Γ-ideals in Γ-AG^{**}-groupoids and regular Γ-AG^{**}-groupoids. Moreover we have proved that the set of Γ-ideals in a regular Γ-AG^{**}-groupoid form a semilattice structure. Also we have characterized a regular Γ-AG^{**}-groupoid in terms of left ideals.
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Characterizations of gamma-AG^{**}-groupoids by the properties their gamma-ideals
In this paper we have discusses Γ-left, Γ-right, Γ-bi-, Γ-quasi-, Γ-interior and Γ-ideals in Γ-AG^{**}-groupoids and regular Γ-AG^{**}-groupoids. Moreover we have proved that the set of Γ-ideals in a regular Γ-AG^{**}-groupoid form a semilattice structure. Also we have characterized a regular Γ-AG^{**}-groupoid in terms of left ideals.
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Γ-Abel-Grassmann's groupoids characterized by their intuitionistic Γ-ideals
In this paper, we have discussed several properties of intuitionistic fuzzy Γ-ideals of a Γ-AG-groupoid which is the generalization of ideals in AG-groupoid We have characterized an intra-regular Γ-AG^{**}-groupoid in terms of intuitionistic fuzzy Γ-left (right, two-sided) ideals, intuitionistic fuzzy (Γ-generalized bi) Γ-bi-ideals, intuitionistic fuzzy Γ-interior ideals and intuitionistic fuzzy Γ-quasi ideals. We have proved that the intuitionistic fuzzy Γ-left (right, interior, quasi) ideals coincide in an intra-regular Γ-AG^{**}-groupoid. We have also shown that the set of intuitionistic fuzzy Γ-two-sided ideals of an intra-regular Γ-AG^{**}-groupoid forms a semilattice structure.
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Characterizations of intra-regular gamma AG-groupoids by the properties of their gamma ideals
We have characterized an intra-regular Γ-AG^{**}-groupoids by using the properties of Γ-ideals (left, right, two-sided ), Γ-interior, Γ-quasi, Γ-bi and Γ-generalized bi and Γ-(1,2)). We have prove that all the Γ-ideals coincides in an intra-regular Γ-AG^{**}-groupoids. It has been examined that all the Γ-ideals of an intra-regular Γ-AG^{**}-groupoids are Γ-idempotent. In this paper we define all Γ-ideals in Γ-AG^{**}-groupoids and we generalize some results.
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On fuzzy-Γ-ideals of Γ-Abel-Grassmann's groupoids
In this paper, we have introduced the notion of Γ-fuzzification in Γ-AG-groupoids which is in fact the generalization of fuzzy AG-groupoids. We have studied several properties of an intra-regular Γ-AG^{**}-groupoids in terms of fuzzy Γ-left (right, two-sided, quasi, interior, generalized bi-, bi-) ideals. We have proved that all fuzzy Γ-ideals coincide in intra-regular Γ-AG^{**}-groupoids. We have also shown that the set of fuzzy Γ-two-sided ideals of an intra-regular Γ-AG^{**}-groupoid forms a semilattice structure.
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On some classes of Abel-Grassmann's 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.
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Intra regular Abel-Grassmann's groupoids characterized by their intuitionistic fuzzy ideals
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.
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Abel-Grassmann's groupoids characterized by (\in ,\in \vee q_{k}) fuzzy bi-ideals
Using the idea of a quasi-coincedece of a fuzzy point with a fuzzy set, the concept of an (α,β)-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.
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Characterizations of right modular groupoids by $(\in, \in \vee q_{k})$-fuzzy ideals
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.
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Characterizations of Abel-Grassmann's groupoids by their intuitionistic fuzzy ideals
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.
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Ideals in Left Almost Semigroups
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.
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Topological Structure on Abel-Grassmann's Groupoids
In this paper we have discussed the ideals in Abel Grassmann's groupoids and construct their topologies.
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Fuzzy Abel Grassmann's Groupoids
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.
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