**Authors:** R. Akhtar, A. Arp, M. Kaminski, J. Van Exel, D. Vernon, C. Washington

### Abstract

A quasigroup identity is said to be of Bol-Moufang type if it involves three variables, two of which occur once on each side and one of which appears twice; moreover, the order in which the variables appear is the same on both sides, and there is only one binary operation in the identity. Answering a question of Drapal, we classify all varieties of quasigroups of Bol-Moufang type where the operation involved is *, /, or \, determining all inclusions among these and providing all necessary counterexamples. This work extends that of Phillips and Vojtechopvsky, who described the relationships among the 26 varieties obtained when the operation is *. We find that 52 varieties, distinct from each other and from the aforementioned 26, are obtained when one allows / or \ as the operation. We determine all inclusions among these varieties, furnishing all necessary counterexamples to complete the classification.

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