Tiling with sets of polyominoes

The definitions and lattice hierarchy previously established for tiling regions with individual polyominoes are extended to finite sets of polyominoes. The problem of tiling the infinite plane with replicas of a finite set of polyominoes is proved to be logically equivalent to Wang's “domino problem,” which is known to be algorithmically undecidable. Several different ways of extending the notion of rep-tility from single polyominoes to sets of polyominoes are discussed. Some related results of Ikeno regarding tiling with polyiamonds (shapes composed of equilateral triangles) are mentioned.