Affiliations: Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan | Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan | Department of Mathematics, Yazd University, Yazd, Iran
Note: [] Corresponding author. Faisal, Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan. E-mails: yousafzaimath@yahoo.com; madadmath@yahoo.com (Madad Khan); shams_ulhaq@yahoo.com (Shamsul Haq); davvaz@yazduni.ac.ir (Bijan Davvaz).
Abstract: Motivated by studying the structural properties of left (right) regular and intra-regular classes of an ordered semigroup, in this paper we have considered these classes in an ordered $\mathcal{AG}$-groupoid and shown that they coincide in a structure of an ordered $\mathcal{AG}$-groupoid with left identity. We have provided some useful connections between ordered $\mathcal{AG}$-groupoids and ordered semigroups. We have proved that every fuzzy right ideal of an ordered $\mathcal{AG}$-groupoid with left identity becomes a fuzzy left ideal but the converse is not valid. Moreover it has shown that a fuzzy left ideal and a fuzzy right ideal coincide in a left regular ordered $\mathcal{AG}$-groupoid with left identity. As an application of our results we get characterizations of left regular ordered $\mathcal{AG}$-groupoids in terms of fuzzy sets.
Keywords: Ordered $\mathcal{AG}$-groupoids, left invertive law, fuzzy ideals
