Ruitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N ≥ 0 such that fN+2 = fN, thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

中文翻译：

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自由 Heyting 代数自同态：Ruitenburg 定理及其他

Ruitenburg 定理说，每一个自同态F一个有限生成的自由 Heyting 代数最终是周期性的，如果F修复所有发电机，但一个。更准确地说，有ñ≥ 0 使得Fñ+2=Fñ，因此周期等于 2。我们使用对偶技术和有界互模拟等级给出了该定理的语义证明。通过相同的技术，我们解决了自由代数的任意自同态的研究。我们表明它们通常不是最终周期性的。然而，当它们是（例如，在局部有限子变体的情况下）时，周期可以明确界定为生成器集的基数的函数。