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List of categories

We give a complete list of all categories occurring in this work. For mostcategories we also indicate where they are discussed. Exceptions arecategories that are so common that nothing needs to be said (such asthe category of sets) and the category of reduced porings (which appearseverywhere in this work). In the notation we use some generic prefixesand suffixes. They have the following meaning:

PrefixesWBIBI(A,P)TO

SufExesRR/NDF

every object has the weak bounded inversion propertyevery object has the bounded inversion propertyevery object is an (A, P)-algebraevery object is totally ordered

the objects are ringsthe objects are reduced ringsthe objects are integral domainsthe objects are fields

almost real closed fields

Example 16.9(e), p. 196; Example 17.8(e), p. 207

(A, P)-algebras of continuous piecewise polyno-mial functions

Example 13.4, p. 163

(A, P)-algebras of continuous piecewise rationalfunctions

(A,P)CPWPFR

(A,P)CPWRFR

Example 21.9(c), p. 240; Theorem 22.7, p. 244; Theorem22.13, p. 250

List of categories

archimedean reduced I-rings

Example 16.3, p. 192; Example 16.4, p. 193

archimedean totally ordered fields

Example 16.8(a), p. 196

continuous piecewise polynomial functions

267

ARCHFR/N

CPWPFR

Example 12.17, p. 156 ff; Example 13.4, p. 163; Example18.7, p. 214; p. 215; p. 218; Example 19.7, p. 223; chapter20, p. 229 ff; Example 21.1, p. 234

continuous piecewise rational functions CPWRFR

Example 12.16, p. 155 f; chapter 21, p. 233 f; chapter 22,p. 244 ff

differentiable semi-algebraic functions

Example 10.20, p. 122; Example 14.11, p. 178

euclidean fields

Example 16.9(b), p. 196; Example 17.8(b), p. 207

euclidean reduced f-rings

Example 12.19, p. 159 ff; Example 13.6, p. 164

f-rings

p. 31 f; p. 219

EFR/N

FR

268

f-rings with l-homomorphisms

p. 31 f; Example 2.2, p. 36

locally ringed spaces with reduced partially or-dered stalks

p. 179 f

locally ringed spaces with strictly real reducedpartially ordered stalks

p.179

List of categories

LFR

POLRSP

SRPOLRSP

partially ordered rings (porings)

chapter 1, p. 22 ff; section 2, p. 35 f; p. 43; p. 64; p. 79;Example 10.9, p. 119; Example 10.10, p. 119; Example10.12, p. 120; Example 10.13, p. 120; Remark 11.14, p. 132;p. 135; p. 189

partially ordered rings with bounded inversion

Example 2.11, p. 41; Example 10.13, p. 120; Example 13.5,p. 164

POR

BIPOR

partially ordered rings with weak bounded inver-sion

Example 2.10, p. 40; Example 10.12, p. 120

preordered (A, P)-algebras

p.91

WBIPOR

(A,P)PREOR

List of categories

preordered rings

269

PREOR

chapter 1, p. 21 ff; section 2, p. 35ff; p. 43; p. 45 f; p. 55 f;p. 64 f; p. 79; Theorem 8.12, p. 89; p. 91; Example 10.9,p. 119; Example 10.12, p. 120; Remark 11.14, p. 132; p. 135;Theorem 12.12, p. 152

preordered rings with bounded inversion

Example 2.11, p. 41

preordered rings with weak bounded inversion

Example 2.10, p. 40; Example 10.12, p. 120

real closed domains

BIPREOR

WBIPREOR

RCD

Proposition 18.1, p. 209; Proof of Theorem 18.3, p. 211;Example 18.6, p. 213

real closed fields

chapter 3, p. 43 ff; Proof of Corollary 16.2, p. 191; Example16.9(a), p. 196; p. 204; Example 17.8(a), p. 207

real closed rings

Example 10.18, p. 122; chapter 12, p. 133 ff; p. 165; chapter15, p. 184; Example 16.5, p. 193; Proposition 18.2, p. 210;chapter 20, p. 229 ff; chapter 21, p. 233 ff; Theorem 22.7,p.244

real rings

p. 22; Example 2.8, p. 39

RCF

RCR

RR

270

reduced i-rings

List of categories

FR/N

p. 32; p. 40; p. 53; Proposition 6.5, p. 60; Remark 10.1,p. 107; Proposition 10.8, p. 117; Example 10.15, p. 120;Example 10.16, p. 121; Example 11.10, p. 130; Example11.12, p. 131; Example 12.17, p. 156; Example 12.19, p. 159;Example 13.3, p. 163; Example 13.5, p. 164; chapter 15,p. 183 ff; Example 16.3, p. 192; Example 16.4, p. 193;p. 209; chapter 19, p. 217 ff

reduced i-rings that are (A, F)-algebras

Example 13.4, p. 163

reduced i-rings with bounded inversion

(A,P)FR/N

BIFR/N

Example 10.15, p. 120; Example 11.10; p. 130; Proof ofProposition 12.4, p. 138; Lemma 12.11, p. 148; Example13.5, p. 164; p. 237; p. 241; Proposition 22.6, p. 244

reduced i-rings with weak bounded inversion

Example 11.10, p. 130

reduced partially ordered (A, F)-algebras

WBIFR/N

(A,P)POR/N

p. 91 f; Proposition 8.14, p. 92; Example 9A.3, p. 95; Proofof Theorem 10.7, p. 116; Example 10.11, p. 119; Example11.9, p. 130; p. 134; Proposition 13.2, p. 163

reduced partially ordered rings POR/N

List of categories

reduced partially ordered rings of Nash functions

Example 10.19, p. 122; p. 179

reduced porings with bounded inversion

271

BIPOR/N

Example 2.11, p. 41; p. 53; Example 98.5, p. 99; Example10.13, p. 120; p. 128; Example 11.10, p. 130; Example 21.1,p.234

reduced porings with weak bounded inversion WBIPOR/N

p. 53; Example 2.10, p. 40; Example 10.12, p. 120; p. 128;Example 11.10, p. 130; p. 165

representable partially ordered rings REPPOR

chapter 6, p. 55 ff; Example 98.5, p. 99; Example 10.14,p. 120; Example 11.10, p. 130; Example 11.12, p. 131

representable partially ordered rings with boundedInVerSIOn

Example 11.10, p. 130

representable partially ordered rings with weakbounded inversion

Example 11.10, p. 130

ringed spaces with totally ordered stalks

p. 220

BIREPPOR

WBIREPPOR

272

rings

root closed totally ordered fields

Example 17.8(d), p. 207

semi-algebraic functions

List of categories

R

SAFR

chapter 7, p. 74 f; Theorem 8.10, p. 89; Theorem 8.12, p. 89;Proof of Proposition 9B.1, p. 98; Example 9C.3, p. 102;Theorem 10.7, p. 116; Corollary 11.5, p. 127; Corollary11.6, p. 128; Proof of Theorem 12.12, p. 152; Example12.15, p. 154; p. 165; Example 16.5, p 193; p. 205

semireal rings

p. 22; p. 133 f

sets

strongly archimedean partially ordered rings

Example 16.4, p. 193

torsion free abelian groups

Example 8.5, p. 82

totally ordered domains

chapter 18, p. 209 ff; section 19, p. 217 ff; Theorem 20.1,p. 229; p. 233

SRR

SETS

TOD

List of categories

totally ordered fields

chapter 3, p. 43 f; Example 8.4, p. 82; chapter 16, p. 189 ff;chapter 17, p. 201 ff; p. 217; p. 233 ff

totally ordered fields with Henselian naturalvaluation

Example 16.9(d), p. 196; p. 208

totally ordered fields of cardinality r;

Example 16.8(b), p. 196

totally ordered integrally closed domains

Example 18.8, p. 214

totally ordered local domains

Example 18.9, p. 214

totally ordered Pythagorean fields

Example 16.9(c), p. 196; Example 17.8(c), p. 207

totally ordered subfields of a fixed totally orderedfield

Example 16.8(c), p. 196

273

TOF

274

von Neumann regular f -rings

List of categories

VNRFR

Example 9C.3, p. 102 ff; Example 10.17, p. 121; p. 125;Proposition 11.4, p. 127; Corollary 11.5, p. 127; Example12.15, p. 154; Example 12.16, p. 155; Example 13.7, p. 165;p. 189; chapter 17, p. 201 ff; p. 217; p. 237; Proposition22.6, p. 244; Theorem 22.7, p. 244; p. 250; Theorem 22.14,p.251

weakly real rings

p. 22 ff; Example 2.8, p. 39

WRR

IndexAIa, totally ordered residue do­

main, 46(A, A+), preordered ring, 21a(a), value of the ring element

a at a point of the realspectrum, 46

absolute property of a reflectivesubcategory, 125

affine real scheme, 179almost real closed field, 197(A, P), preordered ring, 21archimedean f­ring, 192archimedean poring, 192arity of a reflector, 161

bounded inversion property, pre­ordered rings with, 41

Brumfiel spectrum of an f­ring,SpecB(A, P), 49

closed under extremal subob­jects, 82

closed under formation of prod­ucts, 82

closed under strengthening of thepartial order, 108

closure operation, 35compatible function, 141complete ring of functions func­

tor, II, 55complete ring of functions

over a preordered rmg,II(A,P),55

concrete category, 21construct, 21constructible sections, ring of, 63constructible subset of the real

spectrum, 45

constructible topology of the realspectrum, 45

continuous piecewise polynomialfunctions, 156

continuous piecewise rationalfunctions, 155

continuous semi­algebraic func­tions (abstract case),135

continuous semi­algebraic func­tions (geometric case),122

convex ideal, 25co­wellpowered,21

differentiable semi­algebraicfunction, 122

discontinuity, types of, 242

[­extendible object, 85[­injective object, 85elementary equivalence, 44elimination of quantifiers, 44embedding, 28epicomplete object in a category,

75epireflection, 37epireflective subcategory, 37epireflective subcategory gener­

ated by a class of ob­jects, SeP(X), 85

epireflector, 37essential monoreflector, 183euclidean field, 196euclidean f­ring, 159explicit operation, 64extension, 28extension of the base field, 65extremal subobject, 82

276

'P, reduced f-ring reflector, 61factor poring, 25factor porings, existence in a re-

flective subcategory, 107fan in a real spectrum, 170fan in a space of orderings, 170f I{3, canonical homomorphism

between totally orderedresidue domains, 46

F(n), free object over n ele-ments,24

formally real field, 43free functor, 21free object, 21f-ring,31

111Fr(n), 111F:(N), 111F:(n),111F(S), free object over the set S,

24function, 55functional formula, 64functorial extension operator, 37

Harrison topology, 170H-closed reflector, 107H -closed subcategory, 107Hilbert's 17t h problem, 59HRC-field, 197

idempotent functorial extensionoperator, 37

implicit operation, 64injectivity class of E, 86intersection of subobjects, 39inverse topology of a spectral

space, 219irreducible l-ideal, 33

"'a, canonical homomorphisminto r: (0:), 46

Index

"'(0:), totally ordered residuefield, 46

Keimel spectrum, SpeK(A, P),218

'"f / f3' canonical homomorphismbetween totally orderedresidue fields, 46

language of porings, 27lattice of epireflective subcate-

gories, 85lattice of monoreflective subcat-

egories, 88lattice of monoreflectors, 88lattice-ordered ring, 31l-homomorphism, 31l-ideal,32

monoreflection, 37monoreflective subcategory, 37monoreflector, 37

1/, von Neumann regular f -ringreflector, 103

n-ary operation on a reducedporing, 65

n-ary operation on a ring ofsemi-algebraic func-tions, 64

Nash function, 122Nash reflector, 122nontrivial fan in a real spectrum,

170nontrivial fan in a space of order-

ings, 170

II, complete ring of functionsfunctor, 55

1fa, canonical homomorphisminto Alo:, 46

Index

II(A, P), complete ring of func­tions over a preorderedring, 55

partially ordered ring, 22Pierce­Birkhoff Conjecture, 157Pierce­Birkhoff ring, 223podomain, 48poring, 22pos(a), set of positivity, 45positive cone, 21preordered ring, (A, P), (A, A+),

21preordering, 21prime cone in a preordered ring,

45pro constructible subset of the

real spectrum, 45Pw(A), weak order of a weakly

real ring, 22Pw(A), weak preordering of a

ring, 24pythagorean field, 196

quotient poring, 31quotient­closed monoreflective

subcategory, 125

p, real closure reflector, 135PO'.' canonical homomorphism

into p(a), 46r+(A,P), positive cone of the re­

flection r(A, P), 40p (a ), real closed residue field, 46real algebraic numbers, Ro, 43real closed field, 43real closed fields as cogenerating

class, 51real closed local ring, 179real closed residue field, p(a), 46real closed ring, 135real closure, 135

277

real closure of a totally orderedfield, 43

real closure reflector, p, 135real ideal, 22real radical, ­(1(0), 24real ring, 22real spectrum of a preordered

ring, Sper(A, P), 45real spetrum of a ring, Sper(A),

45reflection morphism, 35reflective subcategory, 35reflector, 35relative property of a reflective

subcategory, 126Rep(A, P), representable reflec­

tion,58representable poring, 58Pf//3' canonical homomorphisms

between real closedresidue fields, 46

ring of constructible sections, 63ring of definable functions, 63ring of Nash functions over a

ring, 122ring of semi­algebraic functions,

IT(A, P), 74Ro, real algebraic numbers, 43

IT, semi­algebraic functions re­flector, 74

SAP­field, 170semi­algebraic function (geomet­

ric case), 63semi­algebraic function over a

preordered ring, 65semi­algebraic set, 44semireal ideal, 22semireal ring, 22separating ideal, 224

278

SeP(X), epireflective subcat­egory generated by aclass of objects, 85

(J" (F (n)), semi­algebraic func­tions on RB', 64

sheaf of Nash functions, 179space of orderings, 170SpecB (A, P), Brurnfiel spectrum

of an f ­ring, 49spectral space, 45spectral topology of the real

spectrum, 45SpeK(A, P), Keimel spectrum,

218Sper(A) , real spectrum of a ring,

45Sper(A, P), real spectrum of a

preordered ring, 45Sper(f), functorial map between

real spectra, 45strict real localization, 180strictly real local ring, 179strong amalgamation property of

real closed fields, 43stronger, relation between reflec­

tors, 81strongly archimedean porings,

193strongly cocomplete, 21strongly complete, 21sub­preordered ring, 28subobject, 28subobject generated by a subset,

39subporing, 28supp(a) , support of a prime

cone, 46supp(P) , support of a preorder­

ing, 21support of a preordering P,

Index

supp(P), 21support of a prime cone,

supp(a), 46

theory of porings, 27theory of preordered rings, 27theory of real closed fields, 44theory of reduced porings, 27theory of totally ordered fields,

44total order, 22totally ordered residue domain,

A/a, 46totally ordered residue field,

h;(a), 46

univalence of a formula, 64universal property of a reflection

morphism, 35universal theory, 27

V(a), convex hull of A/a in h;(a),46

V(a), convex hull of A/a inp(a),46

von Neumann regular, f­ring re­flector, u, 103

von Neumann regular f­ring,102

weak bounded inversion, ringswith,40

weak order of a weakly real ring,Pw(A),22

weak preordering of a ring,Pw(A),24

weaker, relation between reflec­tors, 81

weakly real ideal, 22weakly real radical, wy/(O), 23weakly real ring, 22

Index

wellpowered, 21

X -reflective subcategory, 37

279

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