Commutative closed subset

The first section of this chapter is a collection of basic facts about generating sets of closed subsets. We introduce the length function which one obtains from generating subsets, and we establish a connection between generating subsets and commutator subsets. 

In the second section, we introduce the thin residue of a closed subset. The thin residue is a specific commutator subset. We look at the thin residue of a complex product of two closed subsets, at the complex product of the thin residue and the thin radical, and at multiple thin residues of closed subsets. 

In the second section, we shall relate specific closed subsets of S containing a finite closed subset T to the corresponding closed subsets of the quotient scheme of S over T. Among other issues we focus on the relationship between commutators and quotient schemes. This leads naturally to the connection between the thin residue of S and the thin residue of quotient schemes of S. This relationship will be described in Theorem 4.2.8, a result which depends on Lemma 3.2.7. Theorem 4.2.8 turns out to be useful in Section 5.5 where we discuss residually thin schemes. 

In the third chapter of this monograph, we look at closed subsets generated by specific subsets. Our investigation leads us to the definitions of commutator subsets and thin residue of a closed subset, as well as to other characteristic closed subsets of schemes. 

A Lie subalgebra is a subset G of operators of G, which, by itself, is closed with respect to commutation. In other words, the commutator of two elements is a linear combination of the same elements. In mathematical terms,... 

Another definition of interest here involves the subalgebra G of a given Lie algebra G. One has to consider a subset G CG that is closed with respect to the same commutation laws of G,... 

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