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Schwartz N., Madden J.J. — Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings
Schwartz N., Madden J.J. — Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings

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Название: Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings

Авторы: Schwartz N., Madden J.J.


The book lays algebraic foundations for real geometry through a systematic investigation of partially ordered rings of semi-algebraic functions. Real spectra serve as primary geometric objects, the maps between them are determined by rings of functions associated with the spectra. The many different possible choices for these rings of functions are studied via reflections of partially ordered rings. Readers should feel comfortable using basic algebraic and categorical concepts. As motivational background some familiarity with real geometry will be helpful. The book aims at researchers and graduate students with an interest in real algebra and geometry, ordered algebraic structures, topology and rings of continuous functions.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 279

Добавлена в каталог: 18.11.2009

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Предметный указатель
$(A,A^{+})$, preordered ring      21
$a(\alpha)$, value of the ring element a at a point of the real spectrum      46
$A/\alpha$, totally ordered residue domain      46
$f/\beta$, canonical homomorphism between totally ordered residue domains      46
$P_{\omega(A)}$, weak order of a weakly real ring      22
$P_{\omega(A)}$, weak preordering of a ring      24
$r^{+}(A,P)$, positive cone of the reflection $r(A,P)$      40
$R_{0}$, real algebraic numbers      43
$Spec_{B}(A,P)$, Brumfiel spectrum of an f-ring      49
$supp(\alpha)$, support of a prime cone      46
$S_{e}P(X)$, epireflective subcategory generated by a class of objects      85
$V(\alpha)$, convex hull of $A/\alpha$ in $\kappa(\alpha)$      46
$\kappa(\alpha)$, totally ordered residue field      46
$\kappa_{f/\beta}$, canonical homomorphism between totally ordered residue fields      46
$\kappa_{\alpha}$, canonical homomorphism into $\kappa(\alpha)$      46
$\mathcal{F}_{r}(\mathbb{N})$      111
$\mathcal{F}_{r}^{+}(\mathbb{N})$      111
$\nu$, von Neumann regular f-ring reflector      103
$\overline{V}(\alpha)$, convex hull of $A/\alpha$ in $\rho(\alpha)$      46
$\Pi$, complete ring of functions functor      55
$\Pi(A,Р)$, complete ring of functions over a preordered ring      55
$\pi_{\alpha}$, canonical homomorphism into $A/\alpha$      46
$\rho$, real closure reflector      135
$\rho(\alpha)$, real closed residue field      46
$\rho_{f/\beta}$, canonical homomorphisms between real closed residue fields      46
$\rho_{\alpha}$, canonical homomorphism into $\rho(\alpha)$      46
$\sigma$, semi-algebraic functions reflector      74
$\sigma(F(n))$, semi-algebraic functions on $R_{0}^{n}$      64
$\varepsilon$-extendible object      85
$\varepsilon$-injective object      85
$\varphi$, reduced f-ring reflector      61
(A,P), preordered ring      21
Absolute property of a reflective subcategory      125
Affine real scheme      179
Almost real closed field      197
Archimedean f-ring      192
Archimedean poring      192
Arity of a reflector      161
Bounded inversion property, preordered rings with      41
Brumfiel spectrum of an f-ring, ${Spec_{B}}(A,P)$      49
Closed under extremal subobjects      82
Closed under formation of products      82
Closed under strengthening of the partial order      108
Closure operation      35
co-wellpowered      21
Compatible function      141
Complete ring of functions functor, $\Pi$      55
Complete ring of functions over a preordered ring, $\Pi(A,P)$      55
Concrete category      21
Construct      21
Constructible sections, ring of      63
Constructible subset of the real spectrum      45
Constructible topology of the real spectrum      45
Continuous piecewise polynomial functions      156
Continuous piecewise rational functions      155
Continuous semi-algebraic functions (abstract case)      135
Continuous semi-algebraic functions (geometric case)      122
Convex ideal      25
Differentiable semi-algebraic function      122
Discontinuity, types of      242
Elementary equivalence      44
Elimination of quantifiers      44
Embedding      28
Epicomplete object in a category      75
Epireflection      37
Epireflective subcategory      37
Epireflective subcategory generated by a class of objects, $S_{e}P(X)$      85
Epireflector      37
Essential monoreflector      183
Euclidean f-ring      159
Euclidean field      196
Explicit operation      64
Extension      28
Extension of the base field      65
Extremal subobject      82
F(n), free object over n elements      24
F(S), free object over the set S      24
F-ring      31
Factor poring      25
Factor porings, existence in a reflective subcategory      107
Fan in a real spectrum      170
Fan in a space of orderings      170
Formally real field      43
Free functor      21
Free object      21
Function      55
Functional formula      64
Functorial extension operator      37
H-closed reflector      107
H-closed subcategory      107
Harrison topology      170
Hilbert's $17^{th}$ Problem      59
HRC-field      197
Idempotent functorial extension operator      37
Implicit operation      64
Injectivity class of $\varepsilon$      86
Intersection of subobjects      39
Inverse topology of a spectral space      219
Irreducible l-ideal      33
Keimel spectrum, SpeK(A,P)      218
L-homomorphism      31
L-ideal      32
Language of porings      27
Lattice of epireflective subcategories      85
Lattice of monoreflective subcategories      88
Lattice of monoreflectors      88
Lattice-ordered ring      31
Monoreflection      37
Monoreflective subcategory      37
Monoreflector      37
n-ary operation on a reduced poring      65
n-ary operation on a ring of semi-algebraic functions      64
Nash function      122
Nash reflector      122
Nontrivial fan in a real spectrum      170
Nontrivial fan in a space of orderings      170
Partially ordered ring      22
Pierce — Birkhoff Conjecture      157
Pierce — Birkhoff ring      223
Podomain      48
Poring      22
Pos(a), set of positivity      45
Positive cone      21
Preordered ring, (A,P), $(A,A^{+})$      21
Preordering      21
Prime cone in a preordered ring      45
Proconstructible subset of the real spectrum      45
Pythagorean field      196
Quotient poring      31
Quotient-closed monoreflective subcategory      125
Real algebraic numbers, $R_{0}$      43
Real closed field      43
Real closed fields as cogenerating class      51
Real closed local ring      179
Real closed residue field, $\rho(\alpha)$      46
Real closed ring      135
Real closure      135
Real closure of a totally ordered field      43
Real closure reflector, $\rho$      135
Real ideal      22
Real radical, $\sqrt[r]{(0)}$      24
Real ring      22
Real spectrum of a preordered ring, $Sper(A,P)$      45
Real spetrum of a ring, $Sper(A)$      45
Reflection morphism      35
Reflective subcategory      35
Reflector      35
Relative property of a reflective subcategory      126
Rep(A,P), representable reflection      58
Representable poring      58
Ring of constructible sections      63
Ring of definable functions      63
Ring of Nash functions over a ring      122
Ring of semi-algebraic functions, $\sigma(A,P)$      74
SAP-field      170
Semi algebraic function over a preordered ring      65
Semi-algebraic function (geometric case)      63
Semi-algebraic set      44
Semireal ideal      22
Semireal ring      22
Separating ideal      224
Sheaf of Nash functions      179
Space of orderings      170
Spectral space      45
Spectral topology of the real spectrum      45
SpeK(A,P), Keimel spectrum      218
Sper(A), real spectrum of a ring      45
Sper(A,P), real spectrum of a preordered ring      45
Sper(f), functorial map between real spectra      45
Strict real localization      180
Strictly real local ring      179
Strong amalgamation property of real closed fields      43
Stronger, relation between reflectors      81
Strongly archimedean porings      193
Strongly cocomplete      21
Strongly complete      21
Sub-preordered ring      28
Subobject      28
Subobject generated by a subset      39
Subporing      28
supp(P), support of a preordering      21
Support of a preordering P, supp(P)      21
Support of a prime cone, $supp(\alpha)$      46
Theory of porings      27
Theory of preordered rings      27
Theory of real closed fields      44
Theory of reduced porings      27
Theory of totally ordered fields      44
Total order      22
Totally ordered residue domain, $A/\alpha$      46
Totally ordered residue field, $\kappa(\alpha)$      46
Univalence of a formula      64
Universal property of a reflection morphism      35
Universal theory      27
Von Neumann regular f-ring      102
Von Neumann regular, f-ring reflector, $\nu$      103
Weak bounded inversion, rings with      40
Weak order of a weakly real ring, $P_{\omega}(A)$      22
Weak preordering of a ring, $P_{\omega}(A)$      24
Weaker, relation between reflectors      81
Weakly real ideal      22
Weakly real radical, $\sqrt[\omega r]{(0)}}$      23
Weakly real ring      22
Wellpowered      21
X-reflective subcategory      37
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