Atomicity and Well Quasi-Order for Consecutive Orderings on Words and Permutations

Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in infinite posets. The under consideration are (a) being atomic, i.e., not decomposable as a union proper or, equivalently, satisfying the joint embedding property; and (b) well quasi-ordered. posets (1) words over finite alphabet consecutive subword ordering, (2) permutations subpermutation ordering. Underpinning four results characterizations atomicity quasi-order subpath ordering on paths directed graph.