By W. B. Vasantha Kandasamy

ISBN-10: 1931233713

ISBN-13: 9781931233712

More often than not the learn of algebraic constructions bargains with the thoughts like teams, semigroups, groupoids, loops, earrings, 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 spaces.

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

For examples:

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

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 following:

(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 precise right subsets of G such that G = G1 U G2 (G1 now not 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 expounded to be a biring if R = R1 U R2 the place R1 and R2 are right subsets of R and

(R1, +, .) is a ring.

(R2, +, .) is a ring.

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

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**Sample text**

For every p ∈ P and a ∈ A \ {0}; a + p is in A where A ⊂ P. 3 40 P = {0, a, d, b, 1} ⊂ S, P is a S-dual ideal of S. (For A = {0, 1} is a semifield). Similarly we can define Smarandache dual ideal of a semiring. This task is left as an exercise for the reader. 19: Let S be a S-semiring. S is said to be a Smarandache semidivision ring (S-semidivision ring) if the proper subset A ⊂ S is such that iii. iv. A is a S-subsemiring which is non-commutative. A contains a subset P such that P is a semidivision ring that is P is a noncommutative semiring.

57: Let x be an element of R, x is said to be a Smarandache SSelement (SSS-element) of R if there exists y ∈ R \ {x} with x y y = x + y. If a ring has at least one nontrivial SSS-element then we call R a SSS-ring. Now we proceed on to define two more new notions called Smarandache demiring and Smarandache demi module. 58: Let R be a commutative ring with 1. A subset S of R is said to be a Smarandache demi subring (S-demi subring) of R if i. ii. (S, +) is a S-semigroup. (S, y) is a S-semigroup. 59: Let R be a commutative ring with 1.

DEFINITION [46]: Let N be a loop near-ring. An additive subloop A of N is a Nsubloop (right N-subloop) if NA ⊂ A (AN ⊂ A) where NA = {na a ∈ A, n ∈ N}. A non empty subset I of N is called a left ideal in N if i. ii. (L, +) is a normal subloop of (N, +). n(n1 + i) + nr n1 ∈ I for each i ∈ I and n, n1 ∈ N, where nr denotes the unique right inverse of n. A loop near-ring N is said to be left bipotent if Na = Na2 for every a ∈ N. The concepts of center, essential, simple, strictly prime etc can be defined in an analogous way.

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