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.

‎In this paper‎, ‎we introduce a class of higher HOMOMORPHISMs on an algebra $ \mathcal{A} $ and we characterize the structure of them as a linear combination of some ‎sequences‎ of HOMOMORPHISMs‎.‎ ‎Also ‎‎we prove that for any APPROXIMATE higher ring HOMOMORPHISM on a Banach algebra $ \mathcal{A} $ under some sequences of control funtions‎, there exists a unique higher ring HOMOMORPHISM near it. Using special sequences of control functions, we show that the APPROXIMATE higher ring HOMOMORPHISM is an exact higher ring HOMOMORPHISM.

In this paper, we prove that every multipliers on without order Banach algebra A is almost homomorophisms. Also we investigate the continuity of APPROXIMATE multipliers and under special hypothesis we show that each APPROXIMATE multiplier T: A⟶ A is continuous.

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.

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 .

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.

The aim of this paper is to introduce the notion of fuzzy HOMOMORPHISM and fuzzy isomorphism between two n-ary hypergroups and to extend the fuzzy results of fundamental equivalence relations to n-ary hypergroups. We study some of their properties and prove the decomposition theorems for fuzzy HOMOMORPHISM and fuzzy isomorphism.