1 Introduction ................................................. 5
2 Preliminaries ................................................ 6
3 Proof of the main theorem .................................... 8
4 On Condition 4 .............................................. 10
5 On Condition 6 .............................................. 12
6 On Condition 8 .............................................. 15
7 On Condition 9 .............................................. 18
8 The monoid  ................................................ 23
9 The monoid  ................................................ 26
10 The monoids  and  ........................................ 28
11 The monoid  ................................................ 30
12 The monoid  ................................................ 33
13 The monoid  ................................................ 37
14 The monoid  ................................................ 40
15 The monoids  and  ......................................... 42
A. Multiplication tables of monoids of order six ............... 47
References ..................................................... 63
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