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In this paper we introduce a new representation of fuzzy numbers (called the multivalued representation) and a new multiplication on the set F of fuzzy numbers. Introducing a Dorroh type product between fuzzy numbers, we construct some semiring structures on the set F. Two important particular cases of the general Dorroh type product are the cross product, introduced in [Ban, A. I. and Bede, B., Properties of the cross product of fuzzy numbers, Journal of Fuzzy Mathematics, 14 (2006), 513–531] and the Dorroh product, introduced in this paper. An equivalence relation, compatible with the addition and the Dorroh product, is also given.

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