By W. B. Vasantha Kandasamy

ISBN-10: 1931233713

ISBN-13: 9781931233712

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

A entire learn of those bialgebraic buildings and their Smarandache analogues is conducted during this book.

For examples:

A set (S, +, .) with binary operations ‘+’ and '.' is named 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 less 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 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 unique right subsets of G such that G = G1 U G2 (G1 now not 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 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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**Sample text**

A non-empty subset P of S is said to be a Smarandache pseudo right (left) ideal (S- pseudo right (left) ideal) of the semiring S if the following conditions are true. i. ii. e. P⊂ A, A is a semifield in S. For every p ∈ P and every a∈ A; ap ∈ P (pa ∈ P). If P is simultaneously both a S-pseudo right and left ideal we say P is a Smarandache pseudo ideal (S-pseudo ideal). We define Smarandache pseudo dual ideal of S as follows: A non-empty subset P of S is said to be a Smarandache pseudo dual ideal (S-pseudo dual ideal) if the following conditions hold good.

7: Qo is a semifield and Ro is a semivector space over Qo. 8: Let Mn × n = {(aij) aij ∈ Zo}, the set of all n × n matrices with entries from Zo. Clearly Mn × n is a semivector space over Zo. e. 0 < a1 < a2 < … < an-2 < 1). Several interesting properties about semivector spaces can be had from [81 and 122]. The main property which we wish to state about semivector spaces is the concept of basis. 9: A set of vectors (v1, v2, …, vn) in a semivector space V over a semifield S is said to be linearly dependent if there exists a non-trivial relation among them; otherwise the set is said to be linearly independent.

9: A set of vectors (v1, v2, …, vn) in a semivector space V over a semifield S is said to be linearly dependent if there exists a non-trivial relation among them; otherwise the set is said to be linearly independent. (The main difference between vector spaces and semivector spaces is that we do not have negative terms in a semifield over which semivector spaces are built). 10: Let V be a semivector space over the semifield S. For any subset A of V the set of all linear combinations of vectors in A is called the set spanned by A and we shall denote it by 〈A〉.

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