Fuzzy Information and Engineering 2010: Volume I by Lan Shu, Yuebo Zha (auth.), Bing-yuan Cao, Guo-jun Wang,

By Lan Shu, Yuebo Zha (auth.), Bing-yuan Cao, Guo-jun Wang, Si-zong Guo, Shui-li Chen (eds.)

This booklet is the lawsuits of the fifth Annual convention on Fuzzy details and Engineering (ACFIE2010) from Sep. 23-27, 2010 in Huludao, China. This ebook includes 89 papers, divided into 5 major components: In part I, we have now 15 papers on “the mathematical conception of fuzzy systems”. In part II, we have now 15 papers on “fuzzy common sense, platforms and control”. In part III, we now have 24 papers on “fuzzy optimization and decision-making”. In part IV, we now have 17 papers on “fuzzy info, identity and clustering”. In part V, we have now 18 papers on “fuzzy engineering software and tender computing method”.

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X2 ∈L1 So, f (A) is a (λ, μ)-fuzzy ideal of L2 . Theorem 5. Let f : L1 → L2 be a homomorphism and let B be a (λ, μ)-fuzzy sublattice of L2 . Then f −1 (B) is a (λ, μ)-fuzzy sublattice of L1 , where f −1 (B)(x) = B(f (x)), ∀x ∈ L1 . Proof. For any x1 , x2 ∈ L1 , f −1 (B)(x1 ∨ x2 ) ∨ λ = = ≥ = B(f (x1 ∨ x2 )) ∨ λ B(f (x1 ) ∨ f (x2 )) ∨ λ (B(f (x1 )) ∧ B(f (x2 ))) ∧ μ (f −1 (B)(x1 ) ∧ f −1 (B)(x2 )) ∧ μ. Similarly, we have f −1 (B)(x1 ∧ x2 ) ∨ λ ≥ (f −1 (B)(x1 ) ∧ f −1 (B)(x2 )) ∧ μ. So, f −1 (B) is a (λ, μ)-fuzzy sublattice of L1 .

Finally, the implication-based fuzzy left (resp. right) h-ideals of hemirings are considered. Keywords: (∈, ∈ ∨q(λ,μ) )-fuzzy left (resp. right) h-ideals, generalized fuzzy, prime (semiprime), implication-based. 1 Introduction The concept of semirings was introduced by Vandiver in 1935. Ideals of semirings play a central role in the structure theory. However, their properties do not coincide with the usual ring ideals in general which makes people difficult to obtain similar theorems of the usual ring ideals.

8) )- fuzzy h-ideal of N0 . Theorem 3. Let {A}i∈I is a family of (∈, ∈ ∨q(λ,μ) )-fuzzy left h-ideals such that Ai ⊆ Aj or Aj ⊆ Ai for all i, j ∈ I. Then ∪i∈I Ai is an (∈, ∈ ∨q(λ,μ) )fuzzy left h-ideal of R. Theorem 4. Let A be a nonempty subset of a hemiring R. Let B be a fuzzy set in R defined by B(x) = s if x ∈ A, t otherwise, where t < s, 0 ≤ t < μ and λ < s ≤ 1. Then B is an (∈, ∈ ∨q(λ,μ) )-fuzzy left h-ideal of R if and only if A is a left h-ideal of R. Proof. When 0 ≤ t ≤ λ, λ < s < μ, Bα = A λ < α ≤ s, ∅ s < α ≤ μ.

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