Algebra of fuzzy numbers with unimodal membership function

Fuzzy numbers with a unimodal membership function are presented, which have found application in the fuzzy analysis of such subject fields as ecology, chemical technology. In this article, a set of fuzzy numbers with an unimodal membership function of a special type is considered. Two binary operations (addition and multiplication) are given over this set, formulas for calculation are obtained and some properties of this algebra are investigated. It is proved that addition and multiplication are commutative and associative. Moreover, multiplication is distributive over addition. It is shown that there are no neutral and inverse elements over both operations. Note that if we add neutral elements under addition and multiplication to the given algebra, we will obtain a commutative semiring. The conditions under which quasineutral or quasiinverse elements exist are also given.