Bialgebraic Structures

Description

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) ifG1 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.