Bialgebraic Structures and Smarandache Bialgebraic Structures
by W. B. Vasantha Kandasamy

Persian Title: سازه های Bialgebraic و سازه Smarandache Bialgebraic

Summary and Info
Generally the study of algebraic structures deals with the concepts like groups, semigroups, groupoids, loops, rings, nearrings, semirings, and vector spaces. The study of bialgebraic structures deals with the study of bistructures like bigroups, biloops, bigroupoids, bisemigroups, birings, binearrings, 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 (Sbisemigroup) 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 (Sbiloop) if L has a proper subset P which is a bigroup. Let (G, +, .) be a nonempty set. We call G a bigroupoid if G = G1 U G2 and satisfies the following: (G1 , +) is a groupoid (i.e. the operation + is nonassociative). (G2, .) is a semigroup. Let (G, +, .) be a nonempty set with G = G1 U G2, we call G a Smarandache bigroupoid (Sbigroupoid) 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 Sgroupoid. (G2, .) is a Ssemigroup. 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 (Sbiring) (R, +, .) is a nonempty 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 Sring. (R2, +, .) is a Sring.
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