Bialgebraic Structures by W. B. Vasantha Kandasamy

By W. B. Vasantha Kandasamy

Quite often the learn of algebraic buildings offers with the suggestions like teams, semigroups, groupoids, loops, jewelry, near-rings, semirings, and vector areas. The research of bialgebraic buildings offers with the learn of bistructures like bigroups, biloops, bigroupoids, bisemigroups, birings, binear-rings, bisemirings and bivector spaces.
A entire examine of those bialgebraic constructions and their Smarandache analogues is conducted 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 below 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 unique 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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Here W is just a semigroup under ‘+’. Several interesting properties about these structures can be had from [122]. 6 Near-rings and S-near-rings In this section we introduce the concept of near-rings and Smarandache near-rings. As it is essential to know these concepts for building up binear-rings and Smarandache binear-rings, which are very new concepts, we have briefly given them. The study of Smarandache near-rings is very recent; introduced only in the year 2002 [43, 118, 126]. 1: A near-ring is a set N together with two binary operations ‘+’ and 'y' such that i.

N1 → N2 is a near-ring homomorphism; φ need not be even defined on whole of S1. For more about these please refer [118, 126]. Now the concept of Smarandache integral, equiprime and left infra near-rings are defined in an analogous and appropriate ways. These studies are solely assigned to the reader. 49: A Smarandache composition near-ring (S-composition nearring) is a quadruple (C, +, o) where (C, +, y) and C, +, y) are S-near-rings such that (a y b) o c = (a o b) y c for all a, b, c ∈ C. 50: A non zero S-ideal H of G is said to be Smarandache uniform (Suniform) if for each pair of S-ideals K1 and K2 of G such that K1 ∩ K2 = (0); K1 ⊂ H, K2 ⊂ H implies K1 = (0) or K2 = (0).

V. vi. (D, +, 0) is a loop under ‘+’. a + b = 0 implies b + a = 0 for all a, b ∈ D. (D∗, y, 1) is a group where D∗ = D\ {0}. 0 y a = a y 0 = 0 for all a ∈ D. a y (b + c) = a y b + a y c for all a, b, c ∈ D. For every pair a, b ∈ D there exists da, b ∈ D∗ such that for every x ∈ D; a + (b + x) = (a + b) + da, b x. Now [118 and 126] has defined loop near domains analogous to group rings. DEFINITION [118]: Let L be a finite loop under ‘+’ and D be a near domain, the loop near domain DL contains elements generated by di mi where di ∈ D and mi ∈ L where we admit only finite formal sums satisfying the following: i.

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