Cycles in algebraic systems

2. Cycles and their terminology. Let L be an n Xn Latin square or, alternatively, let Q be a quasigroup of order n. Let M be the set of n2 ordered triplets ijk, where k is the entry in the ith row and jth column of L. If S is the set of n2 ordered pairs ij, and if rir: M->S is the projection parallel to the ith coordinate (for example, ir2(ijk) =ik), then for each i, 1_i_3, 7ri is onto S (or equivalently is one-one). There is clearly a one-one correspondence between Latin squares L of order n and sets M of n2 ordered triplets for which the iri are all onto S. Let T: S-S be the involution defined by T(j) = (ji) and Pi: M-M be 7r 'Thrs. For each i, 1 <i<n, let MiCM be the set of triplets which contain i. Each tE Mk, with leading element i, generates what we shall call a cycle on i in the following manner. If 0: Mi-*Mi denotes P2P1P3, the cycle beginning with t shall consist of the triplets