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Yang","submitted_at":"2018-05-05T05:20:27Z","abstract_excerpt":"A differential graded (DG for short) free algebra $\\mathcal{A}$ is a connected cochain DG algebra such that its underlying graded algebra is $$\\mathcal{A}^{\\#}=\\k\\langle x_1,x_2,\\cdots, x_n\\rangle,\\,\\, \\text{with}\\,\\, |x_i|=1,\\,\\, \\forall i\\in \\{1,2,\\cdots, n\\}.$$ We prove that the differential structures on DG free algebras are in one to one correspondence with the set of crisscross ordered $n$-tuples of $n\\times n$ matrixes. We also give a criterion to judge whether two DG free algebras are isomorphic.\n  As an application, we consider the case of $n=2$. 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