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In this paper, we study fuzzy substructures in connection withHv-structures. The original idea comes from geometry, especially from thetwo dimensional Euclidean vector space. Using parameters, we obtain a largenumber of hyperstructures of the group-like or ring-like types. We connect,also, the mentioned hyperstructures with the theta-operations to obtain morestrict hyperstructures, as Hv-groups or Hv-rings (the dual ones).

On a hypergroupoid one can define a topology such that the hyperoperationis pseudocontinuous or continuous. In this paper we extend thisconcepts to the fuzzy case. We give a connection between the classical and thefuzzy (pseudo)continuous hyperoperations.

In a ternary semihyperring, addition is a hyperoperation and multiplicationis a ternary operation. Indeed, the notion of ternary semihyperringsis a generalization of semirings. Our main purpose of this paper is to introducethe notions of fuzzy hyperideal and fuzzy bi-hyperideal in ternary semihyperrings.We give some characterizations of fuzzy hyperideals and investigateseveral kinds of them.

In this note, we introduce the concept of a fuzzy filter of a BLalgebra,with respect to a t-norm briefly, T-fuzzy filters, and give some relatedresults. In particular, we prove Representation Theorem in BL-algebras. Thenwe generalize the notion of a fuzzy congruence (in a BL-algebra) was definedby Lianzhen et al. to a new fuzzy congruence, specially with respect to a tnorm.We prove that there is a correspondence bijection between the set of allT-fuzzy filters of a BL-algebra and the set of all T-fuzzy congruences in thatBL-algebra. Next, we show how T-fuzzy filters induce T-fuzzy congruences,and construct a new BL-algebras, called quotient BL-algebras, and give somehomomorphism theorems.

The operations in the set of fuzzy numbers are usually obtained bythe Zadeh extension principle. But these definitions can have some disadvantagesfor the applications both by an algebraic point of view and by practicalaspects. In fact the Zadeh multiplication is not distributive with respect tothe addition, the shape of fuzzy numbers is not preserved by multiplication,the indeterminateness of the sum is too increasing. Then, for the applicationsin the Natural and Social Sciences it is important to individuate some suitablevariants of the classical addition and multiplication of fuzzy numbers that havenot the previous disadvantage. Here, some possible alternatives to the Zadehoperations are studied.

Let R be a commutative hyperring with identity. We introduceand study prime fuzzy hyperideals of R. We investigate the Zariski topologyon FHspec(R), the spectrum of prime fuzzy hyperideals of R.

In this paper, we deal with Molaei’s generalized groups. We definethe notion of a fuzzy generalized subgroup with respect to a t-norm (orT-fuzzy generalized subgroup) and give some related properties. Especially,we state and prove the Representation Theorem for these fuzzy generalizedsubgroups. Next, using the concept of continuity of t-norms we obtain a correspondencebetween TF(G), the set of all T-fuzzy generalized subgroups of ageneralized group G, and the set of all T-fuzzy generalized subgroups of thecorresponding quotient generalized group. Subsequently, we study the quotientstructure of T-fuzzy generalized subgroups: we define the notion of aT-fuzzy normal generalized subgroup, give some related properties, constructthe quotient generalized group, state and prove the homomorphism theorem.Finally, we study the lattice of T-fuzzy generalized subgroups and prove thatTF(G) is a Heyting algebra.