As an extension of intuitionistic fuzzy sets, we introduce the notion of -fuzzy set ( denoted by SSR) of QS -ideals on a OS -algebra and investigate its properties. Furthermore we study the homomorphic image and inverse image of SSR -fuzzy QS -ideals of a QS -algebra under homomorphism of QS -algebras. Moreover, the Cartesian product of SSR -fuzzy QS -ideals in Cartesian product QS-algebras is given. Finally, novel correlation coefficient between two SSR- fuzzy sets are also studied.

In this paper, we de ne some new homomorphism-monotone parameters for hy-pergraphs. Using these parameters, we extend some graph homomorphism results to hyper-graph case. Also, we present some bounds for some well-known invariants of hypergraphs such as fractional chromatic number, independent numer and some other invariants of hyergraphs, in terms of these parameters.

Picture fuzzy set (PFS) is a novel concept for dealing with uncertainty and a generalization of the traditional fuzzy set (FS) and intuitionistic fuzzy set (IFS) and can easily manage the uncertain nature of human thoughts by incorporating the positive, neutral, negative and refusal membership degrees of an object. Many conceptual ideas on PFSs have been developed so far and applied in diversified fields. In this paper, the concept of picture fuzzy sublattices and picture fuzzy ideals are developed and some of their associated properties are established in detail. Moreover, the sum and product of two picture fuzzy ideals are introduced with their properties. Finally, some properties of picture fuzzy ideals under lattice homomorphism are explored.

Let A be a Banach algebra and (B, B_*) be a dual Banach algebra. A linear map φ : A⟶ B is said to be a δ-homomorphism map if ‖ ‖ φ (a_1 a_2)-φ (a_1)φ (a_2)‖ ‖ ≤ δ ‖ ‖ a_1‖ ‖ ‖ ‖ a_2 ‖ ‖ for every a_1, a_2∈ A. In this paper, we study the δ-homomorphism maps from A into B. Among other things, we prove that if φ : A⟶ B is a δ-homomorphism map and B_* is multiplicative on the algebra generated by φ (A), then φ is bounded and ‖ ‖ φ ‖ ‖ ≤ 1+δ . Let A be a Banach algebra and (B, B_*) be a dual Banach algebra. A linear map φ : A⟶ B is said to be a δ-homomorphism map if ‖ ‖ φ (a_1 a_2)-φ (a_1)φ (a_2)‖ ‖ ≤ δ ‖ ‖ a_1‖ ‖ ‖ ‖ a_2 ‖ ‖ for every a_1, a_2∈ A. In this paper, we study the δ-homomorphism maps from A into B. Among other things, we prove that if φ : A⟶ B is a δ-homomorphism map and B_* is multiplicative on the algebra generated by φ (A), then φ is bounded and ‖ ‖ φ ‖ ‖ ≤ 1+δ .

In this paper we introduce the notion of φ-commutativity for a Banach algebra A, where φ is a continuous homomorphism on A and study the concept of φ-weak amenability for φ-commutative Banach algebras. We give an example to show that the class of φ-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We characterize φ-weak amenability of φ-commutative Banach algebras and prove some hereditary properties. Moreover we verify some of the previous available results about commutative weakly amenable Banach algebras, for φ-commutative φ-weakly amenable Banach algebras.

The clique polynomial of a graph G is the ordinary generating function of the number of complete subgraphs (cliques) of G. In this paper, we introduce a new vertex-weighted version of these polynomials. We also show that these weighted clique polynomials have always a real root provided that the weights are non-negative real numbers. As an application, we obtain a no-homomorphism criteria based on the largest real root of our vertex-weighted clique polynomial.