Series-parallel languages on scattered and countable posets

In this paper, we investigate the recognition by finite automata of languages of countable labelled posets. We unify and generalize several previous results from two different directions: the theory of finite or -free posets, and automata over countable and scattered linear orderings. First, we establish that the smallest class of posets obtained from the empty set and the singleton and closed under finite parallel operation and sequential concatenation indexed by all linear orderings corresponds precisely to the class of scattered and countable N-free posets without infinite antichains. Next, we prove a Kleene-like theorem. We define automata and rational expressions for languages of countable, scattered, N-free labelled posets without infinite antichains, and show that both formalisms have the same expressive power.