Bialgebraic Structures by W. B. Vasantha Kandasamy

By W. B. Vasantha Kandasamy

In general the research of algebraic constructions offers with the techniques 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 spaces.
A entire examine of those bialgebraic buildings and their Smarandache analogues is performed during this book.
For examples:
A set (S, +, .) with binary operations ‘+’ and '.' is named 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 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's 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 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 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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Finally we go on to define Smarandache n-like ring. 65: Let R be a ring, we say R is a Smarandache n-like ring (S-n-like ring) if R has a proper S-subring A of R such (xy)n – xyn –xny – xy = 0 for all x, y ∈ R. This is an identity which is of a special type and hence several interesting results can be determined; the notion of Smarandache power joined and (m, n) power joined are introduced. 66: Let R be a ring. If for every a ∈ A ⊂ R where A is a S-subring there exists b ∈ A such that an = bn for some positive integer m and n then we say R is a Smarandache power joined ring (S-power joined ring).

We say W is a Smarandache normal subring (S-normal subring) if aV = X and Va = Y for all a ∈ R, where both X and Y are S-subrings of R (Here V is a S-subring of R). Almost all relations would follow in a very natural way with simple modifications. 5 Semirings, S-semirings and S-semivector spaces This section is devoted to the introduction of semirings, Smarandache semirings and Smarandache semivector spaces. As the study of bisemirings or bisemivector spaces is very new only being introduced in this book.

V. vi. vii. viii. (DL, +, 0) is a loop under ‘+’. di mi dj mj = di dj mi mj where we assume mi mj ∈ DL as an element and didj , di, dj, dk, dk di ∈ D. mj). mk) where just the elements mi mj mk are juxtaposed and mi, mj, mk, ∈ L. We assume only finite sequence of elements of the form m1 m2 …mk ∈ DL. mi mj = mj mi for all mi mj ∈ L. m1 m2 …mk = m'1 m'2 …m'k ⇔ mi = m'i for i =1, 2, 3, …, k. Since the loop L under ‘+’ and D is also a loop under ‘+’; DL inherits the operation of '+' and '0' serves as the additive identity of the loop DL.

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