Monadic structures over an ordered universal random graph and finite automata

We continue the investigation of the expressive power of the language of predicate logic for finite algebraic systems embedded in infinite systems. This investigation stems from papers of M. A. Taitslin, M. Benedikt and L. Libkin, among others. We study the properties of a finite monadic system which can be expressed by formulae if such a system is embedded in a random graph that is totally ordered in an arbitrary way. The Büchi representation is used to connect monadic structures and formal languages. It is shown that, if one restricts attention to formulae that are $<$-invariant in totally ordered random graphs, then these formulae correspond to finite automata. We show that $=$-invariant formulae expressing the properties of the embedded system itself can express only Boolean combinations of properties of the form ‘the cardinality of an intersection of one-place predicates belongs to one of finitely many fixed finite or infinite arithmetic progressions’.