For a new class of algebras, called EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-algebras, every idempotent element a determines an MV\documentclass[12pt]{minimal}
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\begin{document}$${ MV}$$\end{document}-algebra which is important for the structure of the EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-algebra. Therefore, instead of standard homomorphisms of EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-algebras, we introduce EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-morphisms as a family of MV\documentclass[12pt]{minimal}
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\begin{document}$${ MV}$$\end{document}-homomorphisms from MV\documentclass[12pt]{minimal}
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\begin{document}$${ MV}$$\end{document}-algebras [0, a] into other ones. EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-morphisms enable us to study categories of EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-algebras where objects are EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-algebras and morphisms are special classes of EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-morphisms. The category is closed under product. In addition, we define free EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-algebras on a set X with respect to EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-morphisms. If X is finite, then a free EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-algebra on X is termwise equivalent to the free MV\documentclass[12pt]{minimal}
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\begin{document}$${ MV}$$\end{document}-algebra on X. For an infinite set X, the same is true introducing a so-called weakly free EMV\documentclass[12pt]{minimal}
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\begin{document}$${ EMV}$$\end{document}-algebra.
