In this paper, we define a Cayley–Dickson process for k-coalgebras proving some results that describe the properties of the final coalgebra, knowing the properties of the initial one. Namely, after applying the Cayley–Dickson process for k-coalgebras to a coassociative coalgebra, we obtain a coalternative one. Moreover, the first coalgebra is cocommutative if and only if the final coalgebra is coassociative. Finally we extend these results to a more general approach of D-coalgebras, where D is a k-coalgebra, presenting a class of examples of coalternative (non-coassociative) coalgebras obtained from group D-coalgebras.

中文翻译：

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可替代的代数

在本文中，我们定义了一个 Cayley-Dickson 过程ķ-coalgebras 证明了一些结果，这些结果描述了最终 coalgebra 的性质，知道初始 coalgebra 的性质。即，在应用 Cayley-Dickson 过程后ķ-coalgebras 到一个 coassociative coalgebra，我们得到一个 coalternative。此外，第一个余代数是协交换的当且仅当最后一个余代数是协合的。最后，我们将这些结果扩展到更一般的方法D- 代数，其中D是一个ķ-coalgebra，呈现从组中获得的 coalternative (non-coassociative) coalgebras 的一类示例D-代数。