The paper deals with an algebraic extension of $MV$-algebras based on the definition of generalized Boolean algebras. We introduce a new algebraic structure, not necessarily with a top element, which is called an $EMV$-algebra and every $EMV$-algebra contains an $MV$-algebra. First, we present basic properties of $EMV$-algebras, give some examples, introduce and investigate congruence relations, ideals and filters on this algebra. We show that each $EMV$-algebra can be embedded into an $MV$-algebra and we characterize $EMV$-algebras either as $MV$-algebras or maximal ideals of $MV$-algebras. We study the lattice of ideals of an $EMV$-algebra and prove that any $EMV$-algebra has at least one maximal ideal. We define an $EMV$-clan of fuzzy sets as a special $EMV$-algebra. We show any semisimple $EMV$-algebra is isomorphic to an $EMV$-clan of fuzzy functions on a set. We consider the variety of $EMV$-algebra and we present an equational base for each proper subvariety of the variety of $EMV$-algebras. We establish a categorical equivalencies of the category of proper $EMV$-algebras, the category of $MV$-algebras with a fixed special maximal ideal, and a special category of Abelian unital $\ell$-groups.