Self-similar group action
Suppose is a group and is a finite set which we think of as an alphabet. Let be the set of all words on (viewed as a monoid), and let denote a faithful group action of on as a set (in particular, the action is not an action by monoidal automorphisms). We say that is a self-similar group action if for every and , there exist and such that:
The multiplication by concatenation is happening in the monoid , where it is literally symbol concatenation.
The term self-similar group is typically used for a group that admits a self-similar group action, but in its usual usage, the action is understood to be implicitly specified, even though it is not part of the abstract group structure.