A correspondence between a class of monoids and self-similar group actions. I.
The theory of self-similar group actions arose in the 1980's by considering groups generated by automata [see \textit{R. I. Grigorchuk}, \textit{V. V. Nekrashevich} and \textit{V. I. Sushchanskii}, Automata, dynamical systems and groups, Proc. Steklov Inst. Math. 231, 128-203 (2000); translation from Tr. Mat. Inst. Steklova 231, 134-214 (2000; Zbl 1155.37311)]. It is known that self-similar group actions give rise to monoids. In fact, these semigroups were studied much earlier and were introduced by \textit{J.-F. Perrot} [Contribution à l'étude des monoïdes syntactiques et de certains groupes associés aux automates finis. Université de Paris VI (1972; Zbl 0392.20049)] in the course of generalizing some work of \textit{D. Rees} [Q. J. Math., Oxf. Ser. 19, 101-108 (1948; Zbl 0030.00802)]. A monoid \(S\) is called a left Rees monoid if it satisfies the following conditions: LR1) \(S\) is a left cancellative monoid; LR2) incomparable principal right ideals are disjoint; LR3) each principal right ideal is properly contained in only a finite number of principal right ideals. In sections 2, 3, 4, the author describes Perrot's theory in modern terms. In section 5, he makes the link between Perrot's work and the contemporary theory of self-similar group actions [see \textit{V. V. Nekrashevich}, Self-similar groups. AMS, Providence (2005; Zbl 1087.20032)]. In section 6, the author determines the correspondence between left Rees monoids and the class of \(0\)-bisimple inverse monoids with two properties: i) if \(e\) and \(f\) are idempotents and \(ef\neq 0\), then \(e\) and \(f\) are comparable; ii) there are only a finite number of idempotents above any nonzero idempotent. In section 7, the author shows how left Rees monoids are also related to Mealy machines and double categories.
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