rasul rasuli

In this paper we introduced direct product of intuitionistic fuzzy multigroups of G under norms(IFMSN(G)) and we prove that it will be also IFMSN(G). Next we shall study some important properties and theorems for them. On the other hand we shall give the definition of the identity element, strong upper- lower and weak upper- lowerof them and study the main theorem for this. We shall also give new results on this subject. Also we define the concepts of conjugate and commutative of IFMSN(G) and investigate them under direct product. Finally, we organize them under group homomorphisms and we prove that the image and preimage of direct product of IFMSN(G) will be also IFMSN(G).

In this paper, the concepts of n-monomorphism, n-epimorphism, n-isomorphism, n-equivalent, n-coproduct, n-product, n-injection, n-projection, n-initial object, n-terminal object, n-pushout diagram, n-inverse system, n-inverse limit, and n-homology of categories will be introduced and will be shown the relationship between them. Next, some of their properties and structure characteristics will be investigated and obtained some results about them.

In this paper, we introduce and clarify a new presentation between the n-exact sequence and the n-projective module. Also, we obtain some new results about them

In this paper, we introduce and define the concepts of fuzzy implicative ideals, fuzzy closed implicative ideals and fuzzy commutative ideals of BCI-algebras with respect to T-norms and we investigate some good examples. Next, we present some fundamental properties of them. Also, we link them with implicative ideals, closed implicative ideals and commutative ideals of BCI-algebras such that every fuzzy implicative ideal, fuzzy closed implicative ideals and fuzzy commutative ideals of BCI-algebras with respect to T-norms will be implicative ideals, closed implicative ideals and commutative ideals of BCI-algebras, respectively and we indicate some examples. Later we investigate and generalize the concepts as intersection and cartesian product of them and we show that the intersection and cartesian product of fuzzy implicative ideals, fuzzy closed implicative ideals and fuzzy commutative ideals of BCI-algebras with respect to T-norms are also fuzzy implicative ideals, fuzzy closed implicative ideals and fuzzy commutative ideals of BCI-algebras with respect to T-norms, respectively and characterize their properties of them with examples. Finally, we prove some properties about them under homomorphisms (image and pre-image) of BCI-algebras.

In this paper, we define the conceps of complex fuzzy Lie subalgebras and complex fuzzy ideals of Lie algebras with respect to t-norms and investigate some of characteristics and relationship between them. Next, we introduce the concepos of quotient subalgebras, intersection, sum and direct product of them and prove some results about them. Finally, we introduce and study the image and the inverse image of them under Lie algebra homomorphisms.

In our work in this paper, we define intuitionistic fuzzy complex subgroups with respect to t-norm T and s-norm S and investigate some properties of them in detail. Next, we obtain some results about them and give some relationships between them. Later, we introduce the inverse, composition, intersection and normality of them and we prove some basic new results and present some properties of them such that the inverse and composition of two intuitionistic fuzzy complex subgroups with respect to t-norm T and s-norm S will be intuitionistic complex fuzzy subgroups with respect to t-norm T and s-norm S. Also we consider and give some characterizations of them. Finally, we discuss them under group homomorphisms and investigate some related properties such that the image and preimage of two intuitionistic fuzzy complex subgroups with respect to t-norm T and s-norm S will be intuitionistic complex fuzzy subgroups with respect to t-norm T and s-norm S.

In this paper, by using t-norms, we define complex fuzzy subgroups and normal complex fuzzy subgroups and investigate some of their characteristics of them. Later we introduce and study their intersection and composition of them. Next, we define the concept of normality between two complex fuzzy subgroups under t-norms and obtain some properties of them. Finally, we describe the image and the inverse image of them under group homomorphisms.

The objectives of this article are to present the notions of anti complex fuzzy subalgebras and anti complex fuzzy ideals of Lie algebras under  $S$-norms and we prove that the level subset of them are also subalgebras and ideals of Lie algebras, respectively. Next, we introduce the union and summation of them and we prove that the union and summation of them are also anti complex fuzzy subalgebras and anti complex fuzzy ideals of Lie algebras under  $S$-norms, respectively.  Finally, we investigate the homomorphic image (pre-image)  of them under Lie homomorphisms.

We introduce the concept of Q-soft R- submodules over a commutative ring. Some Properties of Q-soft R- submodules areinvestigated. In particular, we consider properties of intersection and direct sum for Q-soft R- submodules.

In this paper, by using s-norms, the notion of fuzzy bi-ideals in semirings will be defined and investigated some properties of them. Next prime, strongly prime, semiprime, irreducible, strongly irreducible of them will be introduced and obtained some results about them. Latter, they will be investigated under regular and intra-regular semirings. Finally, they will be characterized under totally ordered by inclusion.

In this article, by using norms($T$ and $C$), we present the concept of intuitionistic fuzzy implicative ideals,  intuitionistic fuzzy closed implicative ideals and intuitionistic fuzzy commutative ideals of $BCI$-algebras. Some interesting results of them are given. Characterisations of implicative ideals, closed implicative ideals and commutative ideals of $BCI$-algebras by using them are explored. By using intersections, direct products and homomorphisms, some interesting results are obtained.

In this paper, we use the notion of t-norms to introduce fuzzy subalgebras, fuzzy ideals, fuzzy implicative ideals, fuzzy positive implicative ideals in BCK-algebras. Next, we clarify the links between them and investigate their properties. Finally, we consider them under intersection, cartesian product and homomorphisms(image and pre image) and we study related properties.

In this paper, by using norms (t-norms and t-conorms), we introduce the notions of bifuzzy d-algebras and bifuzzy d-ideals of d-algebras and investigate several interesting properties. Next, we consider their intersection and product. Finally, we obtain some results about them under d-algebra homomorphisms.