Smarandache representation and its applications

Here we for the first time define Smarandache representation of finite S-bisemigroup. We know every S-bisemigroup, S = S1 ∪ S2 contains a bigroup G = G1 ∪ G2. The Smarandache representation S-bisemigroups depends on the S-bigroup G which we choose. Thus this method widens the Smarandache representations. We first define the notion of Smarandache pseudo neutrosophic bisemigroup. Definition 1. Let S = S1 ∪ S2 be a neutrosophic bisemigroup. If S has only bi-group which is not a neutrosophic bigroup, then we all S a Smarandache pseudo neutrosophic bisemigroup(S-pseudo neutrosophic bisemigroup). Example 1. Let S = S1 ∪ S2 where S1 = Q(I) × Q(I) and S2 = {2 × 2 matrices with entries from Q(I)} both S1 and S2 under multiplication is a semigroup. Thus S is a neutrosophic bisemigroup. Take G = G1 ∪ G2 where G1 = {Q\(0) × Q\(0)} and G2 = {set of all 2 × 2 matrices A with entries from Q such that |A | 6=0}. G1 and G2 are groups under multiplication. So S is a pseudo Smarandache Neutrosophic bisemigroup. Now we give the Smarandache representation of finite pseudo Smarandache neutrosophic bisemigroups. Definition 2. Let G = G1 ∪ G2 be a Smarandache neutrosophic bisemigroup and V = V1 ∪ V2 be a bivector space. A Smarandache birepresentation of G on V is a mapping Sρ = S1ρ ∪ S2ρ from H1∪H2 (H1∪H2 is a subbigroup of G which is not a neutrosophic bigroup) to invertible linear bitransformation on V = V1 ∪ V2 such tha