In this paper, we extend the concept of fuzzy subsets given by Zadeh (Inf Control 8:338–353, 1965) to the context of α\documentclass[12pt]{minimal}
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\begin{document}$$^{\alpha }$$\end{document}-fuzzy and α\documentclass[12pt]{minimal}
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\begin{document}$$_{\alpha }$$\end{document}-fuzzy subsets. The aim of this paper is to investigate the concept of α\documentclass[12pt]{minimal}
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\begin{document}$$^{\alpha }$$\end{document}-fuzzy and α\documentclass[12pt]{minimal}
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\begin{document}$$_{\alpha }$$\end{document}-fuzzy subsets in LA\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {L}}}{{\mathcal {A}}}$$\end{document}-semigroups. Some characterizations of (α,β)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha , \beta )$$\end{document}-fuzzy LA\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {L}}}{{\mathcal {A}}}$$\end{document}-subsemigroup, (α,β)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha , \beta )$$\end{document}-fuzzy left, (α,β)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha , \beta )$$\end{document}-fuzzy completely prime and (α,β)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha , \beta )$$\end{document}-fuzzy quasi-prime ideals are obtained. Moreover, we investigate relationships between (α,β)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha , \beta )$$\end{document}-fuzzy completely prime and (α,β)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha , \beta )$$\end{document}-fuzzy quasi-prime ideals of LA\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {L}}}{{\mathcal {A}}}$$\end{document}-semigroups. Finally, we obtain sufficient conditions of an (α,β)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha , \beta )$$\end{document}-fuzzy quasi-prime ideal in order to be an (α,β)\documentclass[12pt]{minimal}
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\begin{document}$$(\alpha , \beta )$$\end{document}-fuzzy completely prime subset.
