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.
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