A Combinatorial Classification of Finite Quasigroups

For a finite groupoid with right cancellation, we define the concepts of a bicycle, of a bicyclic decomposition, and of a bicyclic action of the symmetric group of permutations on a groupoid. An isomorphism criterion based on a bicyclic decomposition gives rise to an effective method for solving problems such as establishing an isomorphism between finite groups with right cancellation, finding their automorphism groups, and listing their subgroupoids. We define an operation of the square of a groupoid using its bicyclic decomposition, which allows one to recognize a quasigroup in a groupoid with right cancellation. On a set of $n$-element quasigroups, we introduce the equivalent relations of being isomorphic and of being of a single type. The factor set of the single-type relation is ordered by an order type relation consistent with squares of quasigroups. A set of $n$-element quasigroups is representable as a union of nonintersecting sequences of quasigroups ordered by a relation of comparison of types of single-type classes that contain them.