Holonomy decomposition of the full transformation semigroup on 5 points

Full transformation semigroups have very regular holonomy decompositions, that can be quite misleading when someone tries to say something about the general case.

x = (2 3 4 5 1), y=(2 2 3 4 5), z=(2 1 3 4 5)

(Generators are in transformation notation, but only the bottom line is displayed. This is not the cyclic notation for permutations.)


(2,S2) wr (3,S3) wr (4,S4) wr (5,S5)

(underscore means that  constant maps are included, Sn is the symmetric group on n points)

Tiling picture:

(The arrows with plugs denote the tile of relation. Numbers on the right denote the hierarchical levels. Nodes are the elements of the extended set of images. Boxes denote subduction equivalence classes. Rectangular nodes are arbitrary chosen class representatives.)

Poset of equivalence class representatives:


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