自由Heyting代数自同态：Ruitenburg定理及其进一步
Free Heyting Algebra Endomorphisms: Ruitenburg's Theorem and Beyond
摘要：
鲁伊滕堡定理说，一个非生成的自由海廷代数的每个自同态f是最终周期的，如果f固定除一个之外的所有生成元。更准确地说，存在N个$\ge$ 0，使得fN +2 = fN，因此周期等于2。我们给出了这个定理的语义证明，使用对偶技术和有界互模拟秩。用同样的方法，我们研究了自由代数之间的任意自同态。我们表明，他们不是，在一般情况下，最终周期。然而，当它们是（例如在局部有限子簇的情况下），周期可以显式地作为生成元集合的基数的函数而有界。
英文摘要：
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 $\ge$ 0 such that f N +2 = f N , 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 between 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.
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