Unary functions

Abstract

We consider $\mathcal{F}$ the class of finite unary functions, $\mathcal{B}$ the class of finite bijections and ${\mathcal{F}}_k$, $k>1$, the class of finite $k-1$ functions. We calculate Ramsey degrees for structures in $\mathcal{F}$ and ${\mathcal{F}}_k$, and we show that $\mathcal{B}$ is a Ramsey class. We prove Ramsey property for the class $\mathcal{OF}$ which contains structures of the form $(A,f,\le )$ where $(A,f)\in \mathcal{F}$ and $\le$is a linear ordering on the set $A$. We also consider a generalization ${\mathcal{M}}_n\mathcal{F}$, $n>1$, of the class $\mathcal{F}$ which contains finite structures of the form $(A,f_1,...,f_n)$ where each $f_i$ is a 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 flows.