# Download Bialgebraic Structures and Smarandache Bialgebraic by W. B. Vasantha Kandasamy PDF

By W. B. Vasantha Kandasamy

More often than not the examine of algebraic buildings offers with the strategies like teams, semigroups, groupoids, loops, jewelry, near-rings, semirings, and vector areas. The examine of bialgebraic constructions 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 ebook.

For examples:

A set (S, +, .) with binary operations ‘+’ and '.' is termed a bisemigroup of style 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 bunch.

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 next:

(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 special right subsets of G such that G = G1 U G2 (G1 no longer integrated in G2 or G2 now not incorporated in G1).

(G1, +) is a S-groupoid.

(G2, .) is a S-semigroup.

A nonempty set (R, +, .) with binary operations ‘+’ and '.' is related 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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Extra resources for Bialgebraic Structures and Smarandache Bialgebraic Structures

Example text

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