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

By W. B. Vasantha Kandasamy

Commonly the examine of algebraic constructions bargains with the suggestions like teams, semigroups, groupoids, loops, jewelry, near-rings, semirings, and vector areas. The learn of bialgebraic constructions offers with the examine 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 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 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 gaggle.

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 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 specified right subsets of G such that G = G1 U G2 (G1 no longer integrated in G2 or G2 no longer integrated 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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