Summary and Info
Generally the study of algebraic structures deals with the concepts like groups, semigroups, groupoids, loops, rings, near-rings, semirings, and vector spaces. The study of bialgebraic structures deals with the study of bistructures like bigroups, biloops, bigroupoids, bisemigroups, birings, binear-rings, bisemirings and bivector spaces. A complete study of these bialgebraic structures and their Smarandache analogues is carried out in this book. For examples: A set (S, +, .) with two binary operations ‘+’ and '.' is called a bisemigroup of type II if there exists two proper subsets S1 and S2 of S such that S = S1 U S2 and (S1, +) is a semigroup. (S2, .) is a semigroup. Let (S, +, .) be a bisemigroup. We call (S, +, .) a Smarandache bisemigroup (S-bisemigroup) if S has a proper subset P such that (P, +, .) is a bigroup under the operations of S. Let (L, +, .) be a non empty set with two binary operations. L is said to be a biloop if L has two nonempty finite proper subsets L1 and L2 of L such that L = L1 U L2 and (L1, +) is a loop. (L2, .) is a loop or a group. Let (L, +, .) be a biloop we call L a Smarandache biloop (S-biloop) if L has a proper subset P which is a bigroup. Let (G, +, .) be a non-empty set. We call G a bigroupoid if G = G1 U G2 and satisfies the following: (G1 , +) is a groupoid (i.e. the operation + is non-associative). (G2, .) is a semigroup. Let (G, +, .) be a non-empty set with G = G1 U G2, we call G a Smarandache bigroupoid (S-bigroupoid) if G1 and G2 are distinct proper subsets of G such that G = G1 U G2 (G1 not included in G2 or G2 not included in G1). (G1, +) is a S-groupoid. (G2, .) is a S-semigroup. A nonempty set (R, +, .) with two binary operations ‘+’ and '.' is said to be a biring if R = R1 U R2 where R1 and R2 are proper subsets of R and (R1, +, .) is a ring. (R2, +, .) is a ring. A Smarandache biring (S-biring) (R, +, .) is a non-empty set with two binary operations ‘+’ and '.' such that R = R1 U R2 where R1 and R2 are proper subsets of R and (R1, +, .) is a S-ring. (R2, +, .) is a S-ring.
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