In this paper, we define the concept of \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}-ary \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-semigroups as a generalization of \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}-ary semigroups and a generalization of \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-semigroups. We prove some results and present many examples of \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}-ary \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-semigroups. Also, we introduce the concept of Green’s equivalence relations on \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}-ary \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-semigroups and prove some properties. Then, the notion of operator \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}-ary semigroup of an \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}-ary \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-semigroup is defined. Finally, we define the notion of \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}-ary \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}-group.
