The definition of period in finite-state Markov chains can be extended to regular languages by considering the transitions of DFAs accepting them. For example, the language $(\Sigma\Sigma)^*$ has period two because the length of a recursion (cycle) in its DFA must be even. This paper shows that the period of a regular language appears as a cyclic group within its syntactic monoid. Specifically, we show that a regular language has period $P$ if and only if its syntactic monoid is isomorphic to a submonoid of a semidirect product between a specific finite monoid and the cyclic group of order $P$. Moreover, we explore the relation between the structure of Markov chains and our result, and apply this relation to the theory of probabilities of languages. We also discuss the Krohn-Rhodes decomposition of finite semigroups, which is strongly linked to our methods.

中文翻译：

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周期性正则语言的半直积分解

通过考虑接受 DFA 的转换，有限状态马尔可夫链中周期的定义可以扩展到常规语言。例如，语言 $(\Sigma\Sigma)^*$ 有周期 2，因为其 DFA 中的递归（循环）长度必须是偶数。本文表明，正则语言的周期在其句法幺半群中表现为循环群。具体来说，我们证明正则语言具有周期 $P$ 当且仅当其句法幺半群同构于特定有限幺半群和阶 $P$ 循环群之间的半直积的子幺半群。此外，我们探索了马尔可夫链的结构与我们的结果之间的关系，并将这种关系应用于语言概率理论。我们还讨论了有限半群的克罗恩-罗德分解，这与我们的方法密切相关。