By W. B. Vasantha Kandasamy
In most cases the learn of algebraic constructions offers with the techniques like teams, semigroups, groupoids, loops, earrings, near-rings, semirings, and vector areas. The learn of bialgebraic buildings bargains with the examine of bistructures like bigroups, biloops, bigroupoids, bisemigroups, birings, binear-rings, bisemirings and bivector areas.
A entire examine of those bialgebraic constructions and their Smarandache analogues is performed during this e-book.
A set (S, +, .) with binary operations ‘+’ and '.' is named a bisemigroup of kind II if there exists right 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 name (S, +, .) a Smarandache bisemigroup (S-bisemigroup) if S has a formal subset P such that (P, +, .) is a bigroup less than the operations of S.
Let (L, +, .) be a non empty set with binary operations. L is expounded to be a biloop if L has nonempty finite right subsets L1 and L2 of L such that L = L1 U L2 and
(L1, +) is a loop.
(L2, .) is a loop or a gaggle.
Let (L, +, .) be a biloop we name L a Smarandache biloop (S-biloop) if L has a formal subset P that is a bigroup.
Let (G, +, .) be a non-empty set. We name G a bigroupoid if G = G1 U G2 and satisfies the subsequent:
(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 name G a Smarandache bigroupoid (S-bigroupoid) if
G1 and G2 are exact right subsets of G such that G = G1 U G2 (G1 no longer incorporated in G2 or G2 now not integrated in G1).
(G1, +) is a S-groupoid.
(G2, .) is a S-semigroup.
A nonempty set (R, +, .) with binary operations ‘+’ and '.' is expounded to be a biring if R = R1 U R2 the place R1 and R2 are right subsets of R and
(R1, +, .) is a hoop.
(R2, +, .) is a hoop.
A Smarandache biring (S-biring) (R, +, .) is a non-empty set with binary operations ‘+’ and '.' such that R = R1 U R2 the place R1 and R2 are right subsets of R and
(R1, +, .) is a S-ring.
(R2, +, .) is a S-ring.
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Bialgebraic Structures and Smarandache Bialgebraic Structures by W. B. Vasantha Kandasamy