Ordinal Notations in Caucal Hierarchy

Caucal hierarchy is a well-known class of graphs with decidable monadic theories. It were proved by L. Braud and A. Carayol that well-orderings in the hierarchy are the well-orderings with order types less than $\varepsilon_0$. Naturally, every well-ordering from the hierarchy could be considered as a constructive system of ordinal notations. In proof theory constructive systems of ordinal notations with fixed systems of cofinal sequences are used for the purposes of classification of provable recursive functions of theories. We show that any well-ordering from the hierarchy could be extended by a monadically definable system of cofinal sequences with Bachmann property. We show that the growth speed of functions from fast-growing hierarchy based on constructive ordinal notations from Caucal hierarchy may be only slightly influenced by the choice of monadically definable systems of cofinal sequences. We show that for ordinals less than $\omega^\omega$ a fast-growing hierarchy based on any system of ordinal notations from Caucal hierarchy coincides with L\"ob-Wainer hierarchy.