  Sets with atoms serve as an alternative to ZFC foundations for mathematics,
where some infinite, though highly symmetric sets, behave in a finitistic way.
Therefore, one can try to carry over analysis of the classical algorithms from
finite structures to some infinite structures. Recent results show that this is
indeed possible and leads to many practical applications. In this paper we
shall take another route to finite analysis of infinite sets, which extends and
sheds more light on sets with atoms. As an application of our theory we give a
characterisation of languages recognized by automata definable in fragments of
first-order logic.
