Faithful Maps

A map X from a subset y of X to X is called faithful if, for any three elements 

The first section of this chapter is a collection of basic facts about faithful maps. 

The second section of this chapter deals with faithfully embedded closed subsets of S. We mainly discuss the question to which extent the property of being faithfully embedded is inherited from given quotient schemes of closed subsets of S to other quotient schemes of closed subsets of S. 

A closed subset of S will be called schurian if it is faithfully embedded in itself. 

Let X be an element in X, and let T be a closed subset of S. It is easy to see that the set of all bijective faithful maps from xT to xT is a group with respect to composition. We call this group the Schur group of T with respect to X. If T = S, we just speak about the Schur group of S.  

Recall that Sch(A) is defined to be the set of all elements a in Aut(A) such that as = Is - Referring to this notation it is clear that the Schur group of S is just the set of all elements ay with a Sch(A). 

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Faithful maps are particularly interesting in connection with constrained sets. Section 6.6 provides a few results about this relationship. They will turn out to be useful in Chapter If. 

Note that faithful maps are necessarily injective. 

Proof, (i) This is just the definition of a faithful map rewritten under the hypothesis that the domain of x contains wt. 

The following lemma shows in which way generating subsets of S and faithful maps work together. 

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