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Some necessary and sufficient conditions for parastrophic invariance of the associative law in quasigroups

Jaiyeola T. G.

Fasciculi Mathematici

2008

Nr 40

25-35

Every quasigroup (S, ⋅) belongs to a set of 6 quasi-groups, 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 πiparastrophe 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}.