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$A_{fin}$ algebra it is $A_\infty$ algebra with $m_n = 0$ for $n >> 0$ and $A^i = 0$ for $|i| >> 0$.

Suppose that we have (compact) dg-algebra $A$, we can build $A_\infty$ minimal model on $H^*(A)$ what we can say about $A$ if we know that minimal model is actually $A_{fin}$?

Or, in another direction, let we have $A_{fin}$ algebra $B$ what we can say about dg-algebra $cobar(bar(B))$?

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