Unary functions

We consider F , the class of …nite unary functions, i.e. the class of pairs of the form (A; f) where A is a …nite set and f is an unary function on A. We also consider two subclasses of F : B and Fk for k > 1. B contains structures (A; f) with the property that f is a bijection of the set A, and Fk contains structures (A; f) with the property that inverse image of every point in A under f has cardinality at most k. In this paper we calculate Ramsey degrees of structures from F and Fk, and we show that B is a Ramsey class. Moreover we introduce various ordered expansions of the classes F and B and we prove Ramsey property for these expansions. In particular we prove Ramsey property for the class OF which contains structures of the form (A; f; ) where (A; f) 2 F and is a linear ordering on the set A. In the case of the class Fk we introduce a precompact expansion with the Ramsey property. We also consider a generalization MnF , n > 1, of the class F which contains …nite structures of the form (A; f1; :::; fn) where each fi is an unary function on the set A. Finally we give a topological interpretation of our results by expanding the list of extremely amenable groups and by calculating various universal minimal ‡ows.