By W. B. Vasantha Kandasamy

ISBN-10: 1931233748

ISBN-13: 9781931233743

The writer stories the Smarandache Fuzzy Algebra, which, like its predecessor Fuzzy Algebra, arose from the necessity to outline buildings that have been extra appropriate with the true global the place the gray parts mattered, not just black or white.

In any human box, a Smarandache n-structure on a collection S capacity a vulnerable constitution {w0} on S such that there exists a series of right subsets Pn–1 integrated in Pn–2 incorporated in … integrated in P2 integrated in P1 integrated in S whose corresponding constructions ensure the chain {wn–1} > {wn–2} > … > {w2} > {w1} > {w0}, the place ‘>’ indicates ‘strictly more suitable’ (i.e. constitution enjoyable extra axioms). This e-book is pertaining to a Smarandache 2-algebraic constitution (two degrees simply of buildings in algebra) on a suite S, i.e. a vulnerable constitution {w0} on S such that there exists a formal subset P of S, that's embedded with an improved constitution {w1}.

Properties of Smarandache fuzzy semigroups, groupoids, loops, bigroupoids, biloops, non-associative jewelry, birings, vector areas, semirings, semivector areas, non-associative semirings, bisemirings, near-rings, non-associative near-rings, and binear-rings are provided within the moment a part of this ebook including examples, solved and unsolved difficulties, and theorems.

Also purposes of Smarandache groupoids, near-rings, and semi-rings in automaton concept, in error-correcting codes, and within the building of S-sub-biautomaton are available within the final bankruptcy.

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**Smarandache Fuzzy Algebra by W. B. Vasantha Kandasamy PDF**

The writer experiences the Smarandache Fuzzy Algebra, which, like its predecessor Fuzzy Algebra, arose from the necessity to outline constructions that have been extra suitable with the genuine international the place the gray parts mattered, not just black or white. In any human box, a Smarandache n-structure on a collection S capability a susceptible constitution {w0} on S such that there exists a series of right subsets Pn–1 incorporated in Pn–2 integrated in … integrated in P2 incorporated in P1 incorporated in S whose corresponding buildings make sure the chain {wn–1} > {wn–2} > … > {w2} > {w1} > {w0}, the place ‘>’ indicates ‘strictly more advantageous’ (i.

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**Extra resources for Smarandache Fuzzy Algebra**

**Example text**

29 Proof: The proof of the following theorem is left for the reader as an exercise. 37: Let µ and λ be fuzzy normal subgroups of a group G and t ∈ [0, 1]. Let A and B be non-empty subsets of the group G. Then µ t ( A ) ⊆ A ⊆ µˆ t ( A ) i. ii. iii. µˆ t ( A ∪ B ) = µˆ t ( A ) ∪ µˆ t ( B ) A ∩ B ≠ φ ⇒ µt ( A ∩ B ) = µt ( A ) ∩ µt ( B ) iv. A ⊆ B ⇒ µt ( A ) µ t ( B ) v. vi. A ⊆ B ⇒ µˆ t ( A ) ∪ µˆ t ( B ) µ t ( A ∪ B ) ⊇ µt ( A ) ∪ µt ( B ) vii. A ∩ B ≠ φ , µˆ t ( A ∩ B ) ⊂ µˆ t ( A ) ∩ µˆ t ( B ) µ ⊂ λ ⇒ µˆ ( A ) ⊆ λˆ ( A ) viii.

If the ring Rµ has no non-zero nilpotent elements, then the fuzzy ideal µ is fuzzy semiprime. THEOREM [99]: A ring R is regular if and only if every fuzzy ideal of R is idempotent. Proof: Refer [99]. 24: A ring R is regular if and only if every fuzzy ideal of R is fuzzy semiprime. Proof: Straightforward hence left for the reader to prove. Now we recall some results on fuzzy subrings from [40]. Now for the time being we assume that R is a commutative ring with unit and M will denote a maximal ideal of R.

If Q' : R' → L is a primary fuzzy ideal of R' then f –1(Q') is a primary fuzzy ideal of R . iii. Let f : R → R' be an epimorphism of rings and Q: R → L and Q' : R' → L be a fuzzy ideals. (a) Q' is primary if and only if f –1(Q') is primary. (b) If Q is constant on ker f, then Q is primary if and only if f (Q) is primary. Now we proceed on to define the notion of weak primary fuzzy ideals. 12: A fuzzy ideal J: R → L is said to be weak primary or in short wprimary if J(xy) = J(x) or J(xy) ≤ J(yn) for some integer n > 0.

### Smarandache Fuzzy Algebra by W. B. Vasantha Kandasamy

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