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

By W. B. Vasantha Kandasamy

In most cases the examine of algebraic constructions bargains with the options like teams, semigroups, groupoids, loops, earrings, 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 learn of those bialgebraic constructions and their Smarandache analogues is performed during this e-book.

For examples:

A set (S, +, .) with binary operations ‘+’ and '.' is termed a bisemigroup of variety 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 lower than the operations of S.

Let (L, +, .) be a non empty set with binary operations. L is related 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's 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 designated right subsets of G such that G = G1 U G2 (G1 no longer incorporated 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.

By W. B. Vasantha Kandasamy

In most cases the examine of algebraic constructions bargains with the options like teams, semigroups, groupoids, loops, earrings, 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 learn of those bialgebraic constructions and their Smarandache analogues is performed during this e-book.

For examples:

A set (S, +, .) with binary operations ‘+’ and '.' is termed a bisemigroup of variety 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 lower than the operations of S.

Let (L, +, .) be a non empty set with binary operations. L is related 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's 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 designated right subsets of G such that G = G1 U G2 (G1 no longer incorporated 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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