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 5 is the set of n2 ordered pairs ij, and if 7r,:Af—>S is the projection parallel to the ith coordinate (for example, ir2(ijk) = ik), then for each i, l^i^3, t< 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 Ti are all onto 5. Let T:S—*S be the involution defined by T(ij) = (ji) and Pi'.M—tM be -k^Ttcí. For each i, 1 ̂ i^n, let MtQM be the set of triplets which contain i. Each ¿£ Mit with leading element i, generates what we shall call a cycle on i in the following manner. If 6 : Mi-^Mi denotes P2PiP3, the cycle beginning with t shall consist of the triplets