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.

Martindale proved that under some conditions every MULTIPLICATIVE isomorphism between two rings is additive. In this paper, we extend this theorem to a larger class of MAPPINGs and conclude that every MULTIPLICATIVE ( ;  )􀀀 derivation is additive.

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.