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

By W. B. Vasantha Kandasamy

ISBN-10: 1931233713

ISBN-13: 9781931233712

Typically the research of algebraic buildings bargains with the innovations like teams, semigroups, groupoids, loops, jewelry, near-rings, semirings, and vector areas. The learn of bialgebraic buildings bargains with the research of bistructures like bigroups, biloops, bigroupoids, bisemigroups, birings, binear-rings, bisemirings and bivector areas.

A whole research of those bialgebraic constructions and their Smarandache analogues is conducted during this e-book.

For examples:

A set (S, +, .) with binary operations ‘+’ and '.' is named 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 below 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 particular right subsets of G such that G = G1 U G2 (G1 now not incorporated in G2 or G2 no longer 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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