By W. B. Vasantha Kandasamy
In most cases the learn of algebraic buildings offers with the suggestions like teams, semigroups, groupoids, loops, jewelry, 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.
A entire examine of those bialgebraic constructions and their Smarandache analogues is conducted during this e-book.
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 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 subsequent:
(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 no longer incorporated in G2 or G2 now not 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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Since the marked arrow has long since become an antiquarian oddity, it would be wrong to claim that it still divides the world into two camps as of old. Nevertheless there is no other teacher that can show so well how our world came to be a perennially divided one. The marked arrow demonstrates what without it would be a mere surmise: that civilization is the issue of a forced union between two fundamentally hostile ways of life, a union which however productive of history has never been a happy one.
110 Thus man has taken another step away from the arrow, but that is only incidental: even the most primitive alteration, the removal of head and feathering, changed the form of the thing almost beyond recognition. It is the function that remains intact. A mere mark or symbol still bestows proprietary right, operating through unlimited time and space, over anything on earth. This is no mere refinement of lawyer’s wit, nor is it a universal human concept: it is rather, as its lineage shows, the hunter’s peculiar idea of property and right.
78 Whatever its origin, the writing of documents was conceived for the same end as the marking of arrows, and the two meet on common ground in the archaic cylinder seal; seal and arrow grew up together, having performed identical functions from the first as instruments of identification and authority. Equipped with such an effective tool, the men of the steppes enjoyed a powerful advantage over the settled agrarians who did not have, because they did not need, anything like it. Against them it was devastating and achieved a permanent conquest; it was an utterly cynical form of persuasion to which they had no answer.
Bialgebraic Structures and Smarandache Bialgebraic Structures by W. B. Vasantha Kandasamy