Bialgebraic Structures and Smarandache Bialgebraic Structures

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. The study of bialgebraic structures started very recently. Till date there are no books solely dealing with bistructures. The study of bigroups was carried out in 1994-1996. Further research on bigroups and fuzzy bigroups was published in 1998. In the year 1999, bivector spaces was introduced. In 2001, concept of free De Morgan bisemigroups and bisemilattices was studied. It is said by Zoltan Esik that these bialgebraic structures like bigroupoids, bisemigroups, binear rings help in the construction of finite machines or finite automaton and semi automaton. The notion of non-associative bialgebraic structures was first introduced in the year 2002. The concept of bialgebraic structures which we define and study are slightly different from the bistructures using category theory of Girard's classical linear logic. We do not approach the bialgebraic structures using category theory or linear logic.