Parastrophic invariance of Smarandache quasigroups

Every quasigroup $(L,\cdot)$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(L,\pi_i)$, $i\in \{1,2,3,4,5,6\}$. It is shown that $(L,\pi_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,\pi_i) \forall i\in \{1,2,3,4,5,6\}$ if and only if for any of some four $j\in \{1,2,3,4,5,6\}$, $(S,\pi_j)$ is an isotope of $(S,\pi_i)$ or $(S,\pi_k)$ for one $k\in \{1,2,3,4,5,6\}$ such that $i\ne j\ne k$. Hence, $(L,\pi_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,\pi_i) \forall i\in \{1,2,3,4,5,6\}$ if and only if any of the six Khalil conditions is true for any of some four of $(S,\pi_i)$.


Introduction
The study of the Smarandache concept in groupoids was initiated by W.B. Vasantha Kandasamy in [18].In her book [16] and first paper [17] on Smarandache concept in loops, she defined a Smarandache loop as a loop with at least a subloop which forms a subgroup under the binary operation of the loop.Here, the study of Smarandache quasigroups is continued after their introduction in Muktibodh [9] and [10].Let L be a non-empty set.Define a binary operation (•) on L : if x • y ∈ L ∀ x, y ∈ L, (L, •) is called a groupoid.If the system of equations ; a • x = b and y • a = b have unique solutions for x and y respectively, then (L, •) is called a quasigroup.Furthermore, if ∃ a !element e ∈ L called the identity element such that ∀ x ∈ L, x • e = e • x = x, (L, •) is called a loop.It can thus be seen clearly that quasigroups lie in between groupoids and loops.So, the Smarandache concept needed to be introduced into them and studied since it has been introduced and studied in groupoids and loops.Definitely, results of the Smarandache concept in groupoids will be true in quasigroup that are Smarandache and these together will be true in Smarandache loops.

Remark 2.1 Definition 2.1 is equivalent to the definition of Smarandache quasigroup in
) is also a loop(and vice versa) while the other adjugates are quasigroups.

Proof
The proof of these follows by using Definition 2.2 and Definition 2.3. ( (2) These ones follow from (1).

Proof
If a quasigroup (L, •) is a SQ, then there exists a subquasigroup S ⊂ L such that (S, •) is associative.According [8], every quasigroup satisfying the associativity law has an identity hence it is a group.So, S is a subgroup of L.
Theorem 2.1 (Khalil Conditions [13]) A quasigroup is an isotope of a group if and only if any one of some six identities are true in the quasigroup.

Main Results
Theorem 3.1 (L, θ) is a Smarandache quasigroup with associative subquasigroup (S, θ) if and only if any of the following equivalent statements is true.
Proof L is a SQ with associative subquasigroup S if and only if s 1 θ(s The proof of the equivalence of ( 1) and ( 2) is as follows.
The proof of the equivalence of ( 3) and ( 4) is as follows.Remark 3.1 In the proof of Theorem 3.1, it can be observed that the isotopisms are triples of the forms (A, I, A) and (I, B, B).All weak associative identities such as the Bol, Moufang and extra identities have been found to be isotopic invariant in loops for any triple of the form (A, B, C) while the central identities have been found to be isotopic invariant only under triples of the forms (A, B, A) and (A, B, B).Since associativity obeys all the Bol-Moufang identities, the observation in the theorem agrees with the latter stated facts.Corollary 3.1 (L, θ) is a Smarandache quasigroup with associative subquasigroup (S, θ) if and only if any of the six Khalil conditions is true for some four parastrophes of (S, θ).

Proof
Let (L, θ) be the quasigroup in consideration.By Lemma 2.1, (S, θ) is a group.Notice that R s Hence, (S, θ * ) is also a group.In Theorem 3.1, two of the parastrophes are isotopes of (S, θ) while the other two are isotopes of (S, θ * ).Since the Khalil conditions are neccessary and sufficient conditions for a quasigroup to be an isotope of a group, then they must be necessarily and sufficiently true in the four quasigroup parastrophes of (S, θ).
(a) R x and L x represent the left and right translation maps in (L, θ) ∀ x ∈ L. (b) R * x and L * x represent the left and right translation maps in (L, θ * ) ∀ x ∈ L. (c) R x and L x represent the left and right translation maps in (L, θ −1 ) ∀ x ∈ L. (d) IR x and IL represent the left and right translation maps in (L, −1 θ) ∀ x ∈ L. (e) R * x and L * x represent the left and right translation maps in (L, (θ −1 ) * ) ∀ x ∈ L. (f) IR * x and IL * represent the left and right translation maps in (L, ( −1 θ)