Zappa-Szép products of free monoids and groups
نویسنده
چکیده
We prove that left cancellative right hereditary monoids satisfying the dedekind height property are precisely the Zappa-Szép products of free monoids and groups. The ‘fundamental’ monoids of this type are in bijective correspondence with faithful self-similar group actions. 2000 AMS Subject Classification: 20M10, 20M50. 1 A class of left cancellative monoids This paper develops some ideas that were touched on first in our paper [6], where we corrected an error in Nivat and Perrot’s [8] generalisation of some pioneering work by David Rees [9]. In this section, we define the class of monoids we shall be interested in. An important role in this paper will be played by free monoids. If X is a set then X denotes the free monoid generated by X. Elements of X are strings and the length of a string x is denoted by |x|. The prefix order on X is defined by x ≤ y iff x = yz for some string z. An S-act or act (X,S) is an action of a monoid S on a set X on the right. If S is a monoid then (S, S) is an act by right multiplication. If Y ⊆ X is a subset such that Y S ⊆ Y then we say that Y is an S-subact or just a subact. Right ideals of S are subacts under right multiplication. If X and Y are acts then a function θ from X to Y is an S-homomorphism or just a homomorphism if θ(xs) = θ(x)s for all x ∈ X and s ∈ S. For a fixed S, we can form the category consisting of S-acts and the homomorphisms between them. The usual definitions from module theory can be adapted to the theory of acts. In particular, we can define when an act is projective. A monoid S is said to be right PP if all its principal
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