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Normally the minimal height function would be used, since this would minimize n. However, we also could use a height function h such that for each 0 < i 5 n there is exactly one equivalence class of height i. 2. Let a , , . . , ar be representatives of the equivalence classes of elements of A of cardinality >1, arranged in such a way that ai < aj implies i < j . Then Observe that by necessity a, = Q. 2 and thus also the Krohn-Rhodes Decomposition Theorem. 1 makes strong use of relational coverings that were introduced in 1,4 but not used until now.

Condition b a implies that a c bt for some t E: S u 1,. By the argument above, we have a = bt and thus a = ast. Since st defines a permutation of a, some power (st)”, n > 1, is the identity on a. Define S = t(st)”-l. Then qsS = q for all q E a. This implies qsSs = qs and therefore q‘is = q’ for all q’ E as = b. 2. 6) all the singletons of Q. In this set A we have the preorder and the equivalence as defined above (for all subsets of 9). 7) Oh = -1. 8) qh = 0 for all singletons q in Q. 9) a b implies ah = bh.