Revisiting a Tiling Hierarchy

We revisit Golomb's tiling hierarchy for finite sets of polyominoes. We consider the case when only translations are allowed for the tiles in a tile set. In this classification, for several levels in Golomb's hierarchy, more types need to be considered. First, we show that there is no general relationship among the types corresponding to a fixed level for the existence of tilings. Second, we review the relationships among the levels from Golomb's hierarchy that are still valid in this setup, and third, we give counterexamples for several relationships in Golomb's hierarchy that fail. As a consequence of the results, we show alternative tiling hierarchies. We also show some unexpected additions to all of the above hierarchies and discuss various implications of tiling capabilities if deficient regions are added to the pool.