# 10. Appendix

Lemma 8   Let be the least right congruence such that is contained in a block B. Then for all we have , i.e.  is contained in the fixing monoid of B.

Proof : 1.11.1
We show that for each we have . Since , we have , and hence as is a right congruence holds. Now implies and we are done.
q.e.d.

1.11

The following lemma is necessary to prove Theorem 6

Lemma 9   For some and we have if and only if .

Proof : 1.11.1
Since and is a right congruence, implies , and as we can conclude .
On the other hand we show that implies by showing by induction on i where . If i=0 we find and hence . Hence let us assume that for all , we have . Now take and hence for some , . The induction hypothesis implies and hence .
q.e.d.

1.11

Proof of Theorem 6: 1.11.1