Highmann's Theorem on Discrete Sets

In this paper we investigate properties of different classes of discrete sets with respect tothe partial-order of subpicture. In particular we take in consideration the classes of convex polyominoesand L-convex polyominoes. In the first part of the paper we study closure properties of theseclasses with respect the order and we give a new characterization of L-convex polyominoes. In thesecond part we pose the question to extend Higman’s theoremto discrete sets. We give a negative answerin the general case and we prove that the set of L-convex polyominoes is well-partially-orderedby using a representation of L-convex polyominoes in terms of words of a regular language.