Our main purpose of this paper is to study the relationship between MULTIPLICATIVE maps and almost MULTIPLICATIVE maps between probabilistic normed algebras. We first derive some properties of invertible elements and their relation with MULTIPLICATIVE maps. Then we show that every complex homomorphism on elements whose probabilistic norm is equal to 1, is bounded. In the following, we give an open problem about the functionally continuous of unital commutative probabilistic Banach algebra. Finally, we prove that every almost MULTIPLICATIVE map that is not a MULTIPLICATIVE map is continuous.

In a MULTIPLICATIVE hyperring, the multiplication is a hyperoperation, while the addition is a binary operation. In this paper, the notion of derivation on MULTIPLICATIVE hyperrings is introduced and some related properties are investigated.

A simple connected graph G of order n  3 is a strongly 2-MULTIPLICATIVE if there is an injective mapping f: V (G)! f1; 2; : : :; ng such that the induced mapping h: A! Z + de ned by h(P) = Q3 i=1 f(vj i ), where j1; j2; j3 2 f1; 2; : : :; ng, and P is the path homotopy class of paths having the vertex set fvj1; vj2; vj3 g, is injective. Let  (n) be the number of distinct path homotopy classes in a strongly 2-MULTIPLICATIVE graph of order n. In this paper we obtain an upper bound and also a lower bound for  (n). Also we prove that triangular ladder, P2 J Cn, Pm J Pn, the graph obtained by duplication of an arbitrary edge by a new vertex in path Pn and the graph obtained by duplicating all vertices by new edges in a path Pn are strongly 2-MULTIPLICATIVE.

In this paper we will give the definition of almost MULTIPLICATIVE linear functionals on commutative Banach algebras and investigate the closeness of these linear functionals with MULTIPLICATIVE linear functionals (AMNM property). َSeveral Banach algebra has this property, but not all of them. Due to relationship between the spectrum of an element of a complex Banach algebra and MULTIPLICATIVE linear functionals. Thus this investigation about almost MULTIPLICATIVE linear functionals give us some information about the spectrum of an element of a Banach algebra. In this paper We will give a very helpful theorem, for this investigation. Also we will show that, if in two commutative Banach algebras A and B, near any almost MULTIPLICATIVE linear functional, there is a MULTIPLICATIVE linear functional, then A×B also has this property and vice versa. We investigate this property for finite product of commutative Banach algebras. Also we will refer to some of the sequential spaces.

Abstract. In this paper, we have introduced and studied the notion of a fuzzy independent pair and obtain some properties of fuzzy α-modular pairs and independent pairs.

For Banach algebras $\mathcal{A}$ and $\mathcal{B}$, we show that if $\mathfrak{A}=\mathcal{A}\times \mathcal{B}$ is unital, then each bi-MULTIPLICATIVE mapping from $\mathfrak{A}$ into a semisimple commutative Banach algebra $\mathcal{D}$ is jointly continuous. This conclusion generalizes  a famous result due to$\check{\text{S}}$ilov, concerning the automatic continuity of homomorphisms between Banach algebras. We also prove that every $n$-bi-MULTIPLICATIVE functionals on $\mathfrak{A}$ is continuous if and only if it is continuous for the case $n=2$.

This paper deals with the automatic continuity of MULTIPLICATIVE polynomial operators on a class of topological algebras. Several results are derived in this direction. We also support our results by some examples.

In this paper, we define the concept of pseudo-irreducible elements in MULTIPLICATIVE lattices and examine its relationship with other important concepts of MULTIPLICATIVE lattices such as prime elements, primary elements, and maximum elements, and then with the help of this concept, we define and analyze the complete comaximal factorization in MULTIPLICATIVE lattices. In particular, we characterize MULTIPLICATIVE lattices whose every element can be written as a product of pseudo-irreducible and pairwise comaximal elements.