On fully regular $$\mathcal{AG }$$-groupoids

One of the best approaches to study one type of algebraic structure is to connect it with other type of algebraic structure which is better explored. In this paper we have accomplished this aim by connecting $${\mathcal{AG }}$$-groupoids with some useful associative and commutative algebraic structures. We have also introduced a fully regular class of an $$ {\mathcal{AG }}$$-groupoid and shown that an $${\mathcal{AG }}$$-groupoid $$\mathcal S $$ with left identity is fully regular if and only if $$\mathcal{L=L }^{i+1}$$, for any left ideal $$\mathcal L $$ of $$\mathcal S $$, where $$i=1,\ldots ,n$$.