Mathematics and Statistics Vol. 8(3), pp. 353 - 362
DOI: 10.13189/ms.2020.080315
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On (2; 2)-regular Non-associative Ordered Semigroups via Its Semilattices and Generated (Generalized Fuzzy) Ideals


Yousef Al-Qudah 1,*, Faisal Yousafzai 2, Mohammed M. Khalaf 3, Mohammad Almousa 1
1 Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman, Jordan
2 Military College of Engineering, National University of Sciences and Technology (NUST), Islamabad, Pakistan
3 Higher Institute of Engineering and Technology, King Mariout, Egypt

ABSTRACT

The main motivation behind this paper is to study some structural properties of a non-associative structure as it hasn't attracted much attention compared to associative structures. In this paper, we introduce the concept of an ordered A*G**-groupoid and provide that this class is more generalized than an ordered AG-groupoid with left identity. We also define the generated left (right) ideals in an ordered A*G**-groupoid and characterize a (2; 2)-regular ordered A*G**-groupoid in terms of these ideals. We then study the structural properties of an ordered A*G**-groupoid in terms of its semilattices, (2; 2)-regular class and generated commutative monoids. Subsequently, compare -fuzzy left/right ideals of an ordered AG-groupoid and respective examples are provided. Relations between an -fuzzy idempotent subsets of an ordered A*G**-groupoid and its -fuzzybi-ideals are discussed. As an application of our results, we get characterizations of (2; 2)-regular ordered A*G**-groupoid in terms of semilattices and -fuzzy left (right) ideals. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works.

KEYWORDS
Ordered AG-groupoid, Non-associativity, Ordered A*G**-groupoid, Left Invertive Law, Generated Ideals and -fuzzy-ideals.

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Yousef Al-Qudah , Faisal Yousafzai , Mohammed M. Khalaf , Mohammad Almousa , "On (2; 2)-regular Non-associative Ordered Semigroups via Its Semilattices and Generated (Generalized Fuzzy) Ideals," Mathematics and Statistics, Vol. 8, No. 3, pp. 353 - 362, 2020. DOI: 10.13189/ms.2020.080315.

(b). APA Format:
Yousef Al-Qudah , Faisal Yousafzai , Mohammed M. Khalaf , Mohammad Almousa (2020). On (2; 2)-regular Non-associative Ordered Semigroups via Its Semilattices and Generated (Generalized Fuzzy) Ideals. Mathematics and Statistics, 8(3), 353 - 362. DOI: 10.13189/ms.2020.080315.