Enumeration of Bi-commutative AG-groupoids

A groupoid satisfying the left invertive law: $ab\cdot c=cb\cdot a$ is called an AG-groupoid and is a generalization of commutative semigroups. We consider the concept of bi-commutativity in AG-groupoids and thus introduce left commutative AG-groupoids, right commutative AG-groupoids and bi-commutative AG-groupoids. We provide a method to test an arbitrary Cayley's table for these AG-groupoids and explore some of the general properties of these AG-groupoids.