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We study these groupoids and obtain some interesting results about them. DEFINITION: Let Zn = {0, 1, 2, ... , n –1} n ≥ 3, n < ∝. Define ∗ a closed binary operation on Zn as follows. For any a, b ∈ Zn define a ∗ b = at + bu (mod n) where (t, u) need not always be relatively prime but t ≠ u and t, u ∈ Zn \ {0}. Clearly {Zn, (t, u), ∗} is a groupoid of order n. For varying values of t and u we get a class of groupoids for the fixed integer n. This new class of groupoids is denoted by Z∗(n). Z∗(n) is an extended class of Z (n) that is Z (n) ⊂ Z∗(n).

Consequent of the example we have the following theorem. 5: P is a left ideal of Zn (t, u) if and only if P is a right ideal of Zn (u, t). Proof: Simple number theoretic computations and the simple translation of rows to columns gives a transformation of Zn (t, u) to Zn (u, t) and this will yield the result. 5: Let Z10 (3, 7) be the groupoid given by the following table: 33 ∗ 0 1 2 3 4 5 6 7 8 9 0 0 3 6 9 2 5 8 1 4 7 1 7 0 3 6 9 2 5 8 1 4 2 4 7 0 3 6 9 2 5 8 1 3 1 4 7 0 3 6 9 2 5 8 4 8 1 4 7 0 3 6 9 2 5 5 5 8 1 4 7 0 3 6 9 2 6 2 5 8 1 4 7 0 3 6 9 7 9 2 5 8 1 4 7 0 3 6 8 6 9 2 5 8 1 4 7 0 3 9 3 6 9 2 5 8 1 4 7 0 This groupoid has no left or right ideals in it.

Justify your answer. 3 Identities in Smarandache Groupoids In this section we introduce some identities in SGs called Bol identity and Moufang identity, P-identity and alternative identity and illustrate them with examples. DEFINITION: A SG (G, ∗) is said to be a Smarandache Moufang groupoid if there exist H ⊂ G such that H is a Smarandache subgroupoid satisfying the Moufang identity (xy)(zx) = (x (yz)) x for all x, y, z in H. DEFINITION: Let S be a SG. If every Smarandache subgroupoid H of S satisfies the Moufang identity for all x, y, z in H then S is a Smarandache strong Moufang groupoid.

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