Parastrophic Invariance of Smarandache Quasigroups * †

Every quasigroup (L, ·) belongs to a set of 6 quasigroups, called parastrophes denoted by (L, πi), i ∈ {1, 2, 3, 4, 5, 6}. It is shown that (L, πi) is a Smarandache quasigroup with associative subquasigroup (S, πi) ∀ i ∈ {1, 2, 3, 4, 5, 6} if and only if for any of some four j ∈ {1, 2, 3, 4, 5, 6}, (S, πj) is an isotope of (S, πi) or (S, πk) for one k ∈ {1, 2, 3, 4, 5, 6} such that i 6= j 6= k. Hence, (L, πi) is a Smarandache quasigroup with associative subquasigroup (S, πi) ∀ i ∈ {1, 2, 3, 4, 5, 6} if and only if any of the six Khalil conditions is true for any of some four of (S, πi).