Theorem 7.2.20. LetS and S1, ..., Sr be as in one of the previous two results, thenS admits a left-zero decomposition (or right-zero respectively). Consequently, the decom- positions in the previous two results are strongG-decompositions.
Proof. We need to show that the decompositions from the previous results are strong de- compositions. It is a strong decomposition since any pairti andtj gives a decomposition
hG, ti, tji \G=hG, tii \G] hG, tji \G, so the result follows from Lemma 7.2.10.
The previous results are describing strong decompositions partitioning either theL- classes orR-classes of a semigroup (as in Figure 7.1); however, a decomposition mixing these turns out to be much more complicated if achievable at all. Different things appear to happen then.
Note that by taking a subgroupHofGwe are able to refine the decompositions even further, as long as no mix of images and kernels occurs (cf. Theorem 4.1.16).
Decompositions and Normalizing Groups
Eventually, in this section the two forms of normalizing groups are considered in context of the decompositions just introduced. Recall, there are
(t1, ..., tr)-normalizing groups and {t1, ..., tr}-normalizing groups.
But before the first results are going to be presented, decompositions of G-closures are necessary.
Definition 7.3.1. LetT ⊆Tn\Snbe a set of transformations with|T| ≥2, andG≤Sn a group.
1. The tuple (G, T) has the decomposable closure (dc) property if for the set
T ={t1, ..., tr}holdshtG1, ..., tGri= r
U
i=1 htGi i.
2. The tuple (G, T) has the strong dc property if for all subsets T0 ⊆ T holds
htG:t ∈T0i= U
t∈T0
htGi.
3. The tuple (G, T) has the weak dc property if for all subsets T0 ⊆ T holds
htG:t ∈T0i= S
t∈T0
htGi.
4. The tuple (G, T) has the very weak dc property if for T = {t1, ..., tr} holds
htG 1, ..., tGri= r S i=1 htG i i.
Like Theorem 7.2.12 on minimal generating sets of strongG-decompositions, the fol- lowing theorem is the corresponding result forG-closures. The proof is almost identical, so it is omitted.
Theorem 7.3.2. Let(G, T)have the strong dc property andS =hTGi, thenT is minimal (among the setsT0 withS =hT0Gi).
Next, we give the first results connecting the new normalization properties from the previous sections.
Lemma 7.3.3. SetT ={t1, ..., tr}and letGbe(t1, ..., tr)-normalizing.
1. If G is a strongly T-decomposing group, then (G, T) has the strong dc property andGis strongly{t1, ..., tr}-normalizing.
2. IfGis aT-decomposing group, then(G, T)has the dc property andGis{t1, ..., tr}- normalizing.
Proof. Consider the following inclusion.
hG, t1, ..., tri \G= r ] i=1 hG, tii \G= r ] i=1 htGi i ⊆ htG1, ..., tGri.
By assumption, it follows that equality holds everywhere.
From the definition it is clear that a strongly {t1, ..., tr}-normalizing group is also
7.3. Decompositions and Normalizing Groups 177
normalizing properties are equivalent. The next result shows that this equivalence is preserved, when dropping the “strongly“ prefix.
Lemma 7.3.4. Let T = {t1, ..., tr} and G be T-decomposing. If G is T-normalizing, thenGis(t1, ..., tr)-normalizing.
Proof. First, assume that whenever there is a word in si andsj lying in hG, tii \G, for
si ∈ hG, tii \Gandsj ∈ hG, tji \G, we write it as a word in the elements ofhG, tii \G.
This is necessary for the following contradiction.
It remains to show thathG, tii \G ⊆ htiGi, for alli. Let Gbe a group with n+ 1
elements and assume there is an element x in hG, tii \G, but not in htGi i. Since G is
T-normalizing,xis a word in t1, tg11, ..., t gn 1 , t2, tg21, ..., t gn 2 , ..., tr, tgr1, ..., t gn r , but not inti, tgi1, ..., t gn
i alone. Moreover, sincehG, tii \GandhG, tji \Gare disjoint, for
i6=j, the elementxis not a word intj, tgj1, ..., t gn
j alone. Hence,xmust be a combination
ofti and sometj. This is a contradiction to the initial assumption, sincehG, Tiis given
by a disjoint union which meansxcannot be such a combination.
From these two results the following consequence is obtained immediately.
Theorem 7.3.5. LetT ={t1, ..., tr}, then the following hold.
1. IfGis (strongly)T-decomposing, thenGis(t1, ..., tr)-normalizing if and only ifG is (strongly){t1, ..., tr}-normalizing.
2. IfGis (strongly) T-decomposing andT-normalizing, then(G, T)has the (strong) dc property.
3. Let(G, T)have the strong dc property, then the following are equivalent: a) Gis stronglyT-decomposing, and
b) Gis(t1, ..., tr)-normalizing andGis{t1, ..., tr}-normalizing. 4. LetT beG-independent. Then, the followinga)andb)are equivalent
a) Gbeing strongly T-decomposing is equivalent to(G, T)having the strong dc property;
b) Gis(t1, ..., tr)-normalizing is equivalent toGis{t1, ..., tr}-normalizing.
Being strongly G-independent is a rather strong and rare property, whence, it is preferable to lessen the conditions and see which of the previous results hold for weak
G-decompositions.
Lemma 7.3.6. LetT ={t1, ..., tr}andGbe weaklyT-decomposing. IfGis(t1, ..., tr)- normalizing, then,
1. (G, T)has the weak dc property and 2. Gis strongly{t1, ..., tr}-normalizing. This also holds when weakening the prefixes. Proof. hG, t1, ..., tri \G= r [ i=1 hG, tii= r [ i=1 htGi i ⊆ htG1, ..., tGri.
Lemma 7.3.7. LetT ={t1, ..., tr}and let the following hold: 1. (G, T)has the weak dc property;
2. Gis weaklyT-decomposing; 3. Gis{t1, ..., tr}-normalizing. Then,Gis(t1, ..., tr)-normalizing.