Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors. They consist of two monoids, the horizontal and the vertical, with an action of the vertical monoid on the horizontal monoid, and a complementary axiom of faithfulness. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. By looking at the syntactic congruence for monoids and as the natural extension in the case of forest algebras, we could define a version of syntactic congruence of a subset of the free forest algebra, not just a forest language. Let [Formula: see text] be a finite alphabet and [Formula: see text] be a pseudovariety of finite forest algebras. A language [Formula: see text] is [Formula: see text]-recognizable if its syntactic forest algebra belongs to [Formula: see text]. Separation is a classical problem in mathematics and computer science. It asks whether, given two sets belonging to some class, it is possible to separate them by another set of a smaller class.Suppose that a forest language [Formula: see text] and a forest [Formula: see text] are given. We want to find if there exists any proof for that [Formula: see text] does not belong to [Formula: see text] just by using [Formula: see text]-recognizable languages, i.e. given such [Formula: see text] and [Formula: see text], if there exists a [Formula: see text]-recognizable language [Formula: see text] which contains [Formula: see text] and does not contain [Formula: see text]. In this paper, we present how one can use profinite forest algebra to separate a forest language and a forest term and also to separate two forest languages.

中文翻译：

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树语言的弱分离问题

森林代数被定义用于研究未排序树的森林 [有序序列] 的语言，其中一个节点可能有两个以上 [有序] 继任者。它们由两个幺半群组成，水平的和垂直的，垂直幺半群对水平幺半群的作用，以及一个互补的忠实公理。伪变量是给定签名的一类有限代数，在取同态图像、子代数和有限直积的情况下是封闭的。通过查看幺半群的句法一致性以及作为森林代数的自然扩展，我们可以定义自由森林代数子集的句法一致性版本，而不仅仅是森林语言。令 [Formula: see text] 为有限字母表， [Formula: see text] 为有限森林代数的伪变种。一种语言[公式：see text] 是 [Formula: see text] - 如果其句法森林代数属于 [Formula: see text] 则可识别。分离是数学和计算机科学中的经典问题。它询问，给定属于某个类的两个集合，是否有可能用另一个更小的类的集合将它们分开。假设给出了森林语言[公式：参见文本]和森林[公式：参见文本]。我们想通过使用[公式：参见文本]-可识别的语言，即给定这样的[公式：参见文本]和[公式：见文本]，如果存在[公式：见文本]-可识别的语言[公式：见文本]，包含[公式：见文本]且不包含[公式：见文本]。在本文中，