一句话：构造按每个排序有限且有限生成的 ω-clone，使其识别非正则无限树语言，推翻简单有限性判定。
在无限字上，ω-半群给出正则语言的良好代数刻画；人们期望无限树也可由某种简单有限代数识别正则语言。本文研究作为无限树代数的多排序结构 ω-clone。作者构造了一个 ω-clone：它在每个排序上有限，并且作为代数有限生成，然而它所识别的语言不是正则无限树语言。这一反例表明，对无限树而言，“被简单有限代数识别”并不等价于正则性；无限树的代数语言理论比 ω-半群情形更复杂，需要更精细的有限性条件或代数不变量。
$\omega$-clones are multi-sorted structures that naturally emerge as algebras for infinite trees, just as $\omega$-semigroups are convenient algebras for infinite words. In the algebraic theory of languages, one hopes that a language is regular if and only if it is recognized by an algebra that is finite in some simple sense. We show that, for infinite trees, the situation is not so simple: there exists an $\omega$-clone that is finite on every sort and finitely generated, but recognizes a non-regular language.
@article{paperbot741,
title = {A non-regular language of infinite trees that is recognizable by a sort-wise finite algebra},
author = {Mikołaj Bojańczyk and Bartek Klin},
journal = {Logical Methods in Computer Science},
volume = {Volume 15, Issue 4},
year = {2019},
doi = {10.23638/lmcs-15(4:11)2019}
}