A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of full binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial. This exhibits an example of a non commutative and non associative 1–affine complete algebra. As far as we know, it is the first example of such an algebra.

中文翻译：

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完全二叉树的代数的仿射完整性

如果对任何同余函数，代数上的一个函数都是同余保留的，它会将成对的一致元素映射到成对的同等元素上。我们证明，在全二叉树的代数上，其叶子由至少包含三个字母的字母标记，当且仅当它是多项式时，该函数才是全等的。这展示了一个非可交换且非关联的1-仿射完全代数的例子。据我们所知，这是此类代数的第一个例子。