We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff–Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a \(\mathbin {\cup _1}\)-product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense of Munkholm.

中文翻译：

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同伦Gerstenhaber代数是强同伦可交换的

我们证明，从Stasheff–Halperin的意义上讲，任何同伦Gerstenhaber代数自然都是强同伦可交换（shc）代数，具有同伦关联结构图。在杆构造上存在与\（\ mathbin {\ cup _1} \）乘积相对应的某些附加运算时，结构图变为同伦可交换的，因此可以得到Munkholm的shc代数。