Closed Subsets Generated by Involutions

Since T is 2-valenced, our hypothesis that 1 = 0 T) forces T to have odd valency. Thus, by Lemma 2.3.6(ii), 0 T) has odd valency, so that our claim follows from (i). 

In this section, the letter L stands for a set of involutions of S. We shall look at (L). Instead of we shall write . 

The claim is obviously true if i = 0. Therefore, we assume that i G 1. n. Then, assuming that Sj i G we obtain from Si G Si L that Sj G 

Since r G pq and q Gtu, r G ptu. Thus, there exists an element s in pt such that r su. 

As a consequence of the remark right before Lemma 3.4.2 we obtain that, in both of these definitions, the equation (r) = (p) + q) can be replaced with the condition that (p) + (q) (r). 

Lemma 3.4.6 Let p and q be elements in (L such that p G S (q)- Then the following hold. 

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Generating sets of elements of S are particularly interesting if they consist of involutions. In fact, a major part of this monograph deals with closed subsets generated by distinguished sets of involutions. Section 3.4 deals with general aspects of closed subsets generated by involutions. 

Some of the results in the last two sections of this chapter, in particular Theorem 3.6.4 and Theorem 3.6.6, foreshadow the importance of closed subsets generated by involutions in scheme theory. 

In this chapter, we deal only with closed subsets generated by a set of two involutions. However, our approach indicates that the concepts and techniques developed in this chapter generalize meaningfully to closed subsets generated by an arbitrary finite set of involutions. 

Closed subsets generated by sets of involutions turn out to be an interesting subject, especially if one imposes appropriate extra conditions on the set of the generating involutions. As an example, we introduce constrained sets of involutions as another example, we look at sets of involutions satisfying the exchange condition. A constrained set of involutions which satisfies the exchange condition will be called a Coxeter set. 

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