Quotient Schemes

A subset R of S is called naturally valenced if each element of R has finite valency. The present chapter starts with the observation that naturally valenced schemes give rise to quotient schemes over finite closed subsets. After the definition of quotient schemes we shall always assume S to be naturally valenced. 

In the first section, we compute the structure constants of quotient schemes of S in terms of those of S. We relate the complex multiplication in S to the one in quotient schemes of S, and we look at the relationship between subschemes of quotient schemes and quotient schemes of subschemes. 

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 section, we assume S to have finite valency. We investigate the arithmetic between the structure constants of S and the structure constants of the quotient schemes of S. 

For each finite closed subset T of S, we call S//T the quotient scheme of S over T. 

The following lemma relates the complex multiplication in S with the complex multiplication in quotient schemes of S over closed subsets of finite valency. 

The following result describes the relationship between quotient schemes of subschemes of S and subschemes of quotient schemes of S. Note that its first part generalizes Theorem 4.1.3(i). 

Let us now look how the thin residue works together with quotient schemes. 

From Theorem 3.2.1 (i) we know that O U J ) is strongly normal in T. Thus, by Lemma 4.2.5(h), T//0 T) is thin. Thus, we obtain from Lemma 4.2.7(i) that the thin residue 0 S(T) of T is the uniquely defined smallest closed subset of S having a thin quotient scheme. 

Lemma 5.6.2 and Lemma 5.6.3(ii) say that the property of being residually thin is inherited by closed subsets and by quotient schemes over normal closed subsets. Here is a partial converse. 

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. 

The Schur group is the subject of the third section of this chapter. We shall see, in particular, that S is schurian if and only if S is isomorphic to a quotient scheme of a thin scheme, and this is the case if and only if, modulo the group... 

The following two propositions show that the property of being faithfully embedded is inherited to closed subsets and to quotient schemes. 

Corollary 6.3.2 The scheme S is schurian if and only if S is a quotient scheme of a thin scheme. 

Quotient schemes are introduced in the fourth chapter of this monograph. Factorization over non-normal closed subsets provides us with a particularly smooth approach to a generalization of Ludwig Sylow s theorems on finite... 

Morphisms are related to faithful maps, which lead naturally to the notion of a faithfully embedded closed subset. Such subsets provide an appropriate language for an attempt to establish so-called recognition theorems. These theorems deal with the question of which schemes are quotient schemes of thin schemes. We shall come back to recognition theorems and their role in scheme theory later in this preface. 

It is well known (and easy to see) that schemes are not necessarily isomorphic to quotient schemes of thin schemes. However, it seems that a general scheme theoretic characterization of such quotient schemes is out of reach. It is for this reason that one might instead ask for specific sufficient conditions for a scheme to be isomorphic to a quotient scheme of a thin scheme. 

Although activation of the AHR by DLCs is a key event, mechanistic data indicate that AHR-mediated responses are not well conserved across species, with lower sensitivity in humans. A TEF value for a DLC based on rodent data may overestimate the potency of a DLC in humans, and this has not been considered in the current risk assessment of DLCs. Thus, the current TEF-Toxic Equivalency Quotient scheme tends to compound the conservative estimates of risk that exist within standard risk assessment approaches. Moreover mechanistic differences will now be considered by US EPA in the risk assessment of chemical carcinogens. The mechanistic data currently available for receptor-mediated DLCs and PPs clearly indicate that humans respond differently to these two classes of rodent carcinogens, and these data will need to be incorporated into cancer risk assessments for these chemicals. Full appreciation of the species differences in these receptor mechanisms will require continued development and refinement of models such as primary... 

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