Some Necessary and Sufficient Conditions for Parastrophic Invariance of the Associative Law in Quasigroups

Every quasigroup (S, ·) belongs to a set of 6 quasigroups, called parastrophes denoted by (S, πi), i ∈ {1, 2, 3, 4, 5, 6}. It is shown that isotopy-isomorphy is a necessary and sufficient condition for any two distinct quasigroups (S, πi) and (S, πj), i, j ∈ {1, 2, 3, 4, 5, 6} to be parastrophic invariant relative to the associative law. In addition, a necessary and sufficient condition for any two distinct quasigroups (S, πi) and (S, πj), i, j ∈ {1, 2, 3, 4, 5, 6} to be parastrophic invariant under the associative law is either if the πi-parastrophe of H is equivalent to the πi-parastrophe of the holomorph of the πi-parastrophe of S or if the πi-parastrophe of H is equivalent to the πk-parastrophe of the πi-parastrophe of the holomorph of the πi-parastrophe of S, for a particular k ∈ {1, 2, 3, 4, 5, 6}.