# Special Groups

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*M. A. Dickmann; F. Miraglia*

This monograph presents a systematic study of Special
Groups, a
first-order universal-existential axiomatization of the theory of quadratic
forms, which comprises the usual theory over fields of characteristic
different from 2, and is dual to the theory of abstract order spaces.

The heart of our theory begins in Chapter 4 with the result that Boolean
algebras have a natural structure of reduced special group. More deeply,
every such group is canonically and functorially embedded in a certain
Boolean algebra, its Boolean hull. This hull contains a wealth of
information about the structure of the given special group, and much of
the later work consists in unveiling it.

Thus, in Chapter 7 we introduce two series of invariants “living”
in the Boolean hull, which characterize the isometry of forms in any reduced
special group. While the multiplicative series—expressed in terms of meet
and symmetric difference—constitutes a Boolean version of the
Stiefel-Whitney invariants, the additive series—expressed in terms of
meet and join—, which we call Horn-Tarski invariants, does not
have a known analog in the field case; however, the latter have a considerably
more regular behaviour. We give explicit formulas connecting both series, and
compute explicitly the invariants for Pfister forms and their linear
combinations.

In Chapter 9 we combine Boolean-theoretic methods with techniques from
Galois cohomology and a result of Voevodsky to obtain an affirmative
solution to a long standing conjecture of Marshall concerning quadratic
forms over formally real Pythagorean fields.

Boolean methods are put to work in Chapter 10 to obtain information
about categories of special groups, reduced or not. And again in Chapter
11 to initiate the model-theoretic study of the first-order theory of
reduced special groups, where, amongst other things we determine its
model-companion.

The first-order approach is also present in the study of some outstanding
classes of morphisms carried out in Chapter 5, e.g., the pure embeddings
of special groups. Chapter 6 is devoted to the study of special groups of
continuous functions.

#### Table of Contents

# Table of Contents

## Special Groups

- Contents vii8 free
- Preface xi12 free
- Chapter 1. Special Groups 118 free
- Chapter 2. Pfister Forms, Saturated Subgroups and Quotients 3350
- Chapter 3. The Space of Orders of a Reduced Group. Duality 5067
- Chapter 4. Boolean Algebras and Reduced Special Groups 5976
- Chapter 5. Embeddings 7491
- Chapter 6. Special Groups of Continuous Functions 100117
- Chapter 7. Horn-Tarski and Stiefel-Whitney Invariants 135152
- Chapter 8. Algebraic K-theory of Fields and Special Groups 173190
- Chapter 9. Marshall's Conjecture for Pythagorean Fields 182199
- Chapter 10. The category of special groups 209226
- Chapter 11. Some Model Theory of Special Groups 217234
- Appendix A. The Universal Theory of Reduced Special Groups 231248
- Appendix B. Table of References for [DM1] and [DM2] 237254
- Bibliography 239256
- Index 242259