Fans in the theory of real semigroups
M. Dickmann
CNRS
Institut de Mathématiques de Jussieu
Universités Paris 6 et 7
Paris – France
XIX Coloquio Latinoamericano de Algebra, Pucón, Chile,
Diciembre 2012
(Joint work with A. Petrovich, Universidad de Buenos Aires, Argentina.)
Ternary semigroups.
Definition 1
A ternary semigroup (abbreviated TS) is a structure
hS, · , 1, 0, −1i with individual constants 1, 0, −1, and a binary
operation “ · ” such that:
[TS1] hS, · , 1i is a commutative semigroup with unit.
[TS2] x 3 = x for all x ∈ S.
[TS3] −1 6= 1 and (−1)(−1) = 1.
[TS4] x · 0 = 0 for all x ∈ S.
[TS5] For all x ∈ S, x = −1 · x ⇒ x = 0.
We shall write −x for −1 · x.
2
Ternary semigroups.
Definition 1
A ternary semigroup (abbreviated TS) is a structure
hS, · , 1, 0, −1i with individual constants 1, 0, −1, and a binary
operation “ · ” such that:
[TS1] hS, · , 1i is a commutative semigroup with unit.
[TS2] x 3 = x for all x ∈ S.
[TS3] −1 6= 1 and (−1)(−1) = 1.
[TS4] x · 0 = 0 for all x ∈ S.
[TS5] For all x ∈ S, x = −1 · x ⇒ x = 0.
We shall write −x for −1 · x.
2
Examples 2
(a) The three-element structure 3 = {1, 0, −1} has an obvious
ternary semigroup structure.
(b) For any set X , the set 3X under pointwise operation and
constant functions with values 1, 0, −1, is a TS.
2
M. Dickmann, A. Petrovich, Real Semigroups and Abstract Real Spectra.
I, Contemporary Math. 344 (2004), 99-119, Amer. Math. Soc.
Real semigroups.
To define the notion of real semigroup one adds a ternary
relation, D, to the language of ternary semigroups, and 8 or 9
simple axioms governing its behaviour. I’ll omit them. The
intended meaning of D(a, b, c) —hereafter denoted c ∈ D(a, b)—
is: “the element c 6= 0 (of a ring or a related structure) is
represented by the binary quadratic form aX 2 + bY 2 ”, i.e., it is a
value of this polynomial for some of the arguments X , Y (possibly
in some subset). For example, if A is a (commutative, unitary) ring
and T ⊆A is a preorder of A (i.e., T + T ⊆T , T · T ⊆T , A2 ⊆T and
−1 6∈ T ), for a, b, c ∈ A \ {0},
c ∈ D(a, b) ⇔ ∃ s, t ∈ T such that c = as + bt.
Here are the fundamental examples:
The real semigroup associated to a ring.
Reminder. (The real spectrum of a ring)
The real spectrum of a (commutative, unitary) ring, A is denoted
by Sper (A). The basic theory of the real spectrum is expounded
in:
[BCR] J. Bochnak, M. Coste, M.F. Roy, Real Algebraic Geometry, Springer-Verlag,
Berlin, Heidelberg, New York (1998). [English version of Géométrie algébrique réelle,
Ergeb. Math. 12, Springer-Verlag, Berlin, Heidelberg, New York (1987).] (§ 7.1).
[D] M. Dickmann, Applications of model theory to real algebraic geometry; a survey,
Lecture Notes Math. 1130, Springer-Verlag (1985).
[DST] M. Dickmann, N. Schwartz, M. Tressl, Spectral Spaces (in preparation). (§ 23)
[KS] M. Knebusch, C. Scheiderer, Einführung in der Reelle Algebra, Vieweg,
Braunschweig, 1989, x + 189 pp. (Ch. 3)
where one can find a discussion of the geometric motivation of this
construction.
For any semi-real ring A, the set GA consists of all functions
a : Sper (A) → 3, for a ∈ A, where

if a ∈ α \ (−α)
 1
0
if a ∈ α ∩ −α
a (α) =

−1 if a ∈ (−α) \ α.
with the operation induced by product in A. GA is a ternary
semigroup. More generally, given a (proper) preorder T of a ring A
one can relativize the definition above to T , by considering
functions a defined on Sper (A, T ) = {α ∈ Sper (A) | α ⊇ T },
instead of Sper (A); the corresponding ternary semigroup
P 2will be
denoted GA,T . The case above is obtained for T =
A .
2
The real semigroup structure of GA
The ternary semigroup GA endowed with the ternary relation
[R] c ∈ DA (a, b) ⇔ ∀α ∈ Sper (A) [c(α) = 0 ∨ a(α)c(α) = 1
∨b(α)c(α) = 1],
for a, b, c ∈ A, is a real semigroup. A similar definition with
SpecR (A) replaced by SpecR (A, T ) —T a proper preorder of A—
also endows the ternary semigroup GA,T , with a structure of real
semigroup.
The real semigroup structure of GA
The ternary semigroup GA endowed with the ternary relation
[R] c ∈ DA (a, b) ⇔ ∀α ∈ Sper (A) [c(α) = 0 ∨ a(α)c(α) = 1
∨b(α)c(α) = 1],
for a, b, c ∈ A, is a real semigroup. A similar definition with
SpecR (A) replaced by SpecR (A, T ) —T a proper preorder of A—
also endows the ternary semigroup GA,T , with a structure of real
semigroup.
We remark that GA also carries a second ternary relation —called
transversal representation— given by
[TR] c ∈ D t (a, b) ⇔ ∀α ∈ Sper(A)[(c(α) = 0 ∧ a(α) = −b(α))
A
∨ a(α)c(α) = 1 ∨ b(α)c(α) = 1].
It should be noted that, though the multiplication of A is preserved
in passing to its associated RS, GA , addition is destroyed. However,
transversal representation codes what remains of the sum of A in
passing to GA . Each of these representation relations is
(quantifier-free) definable in terms of the other (using the ternary
semigroup operations and constants).
The following references contain a comprehensive treatment of the
theory of real semigroups:
[DP1] M. Dickmann, A. Petrovich, Real Semigroups and Abstract Real Spectra. I,
Contemporary Math. 344 (2004), 99-119, Amer. Math. Soc.
[DP2] M. Dickmann, A. Petrovich, The Three-Valued Logic of Quadratic Form Theory
over Real Rings, in “Andrzej Mostowski and Foundational Studies” (A. Ehrenfeucht,
V. W. Marek, M. Srebrny, eds.), IOS Press, Amsterdam, 2008, 49-67.
[DP3] M. Dickmann, A. Petrovich, Spectral Real Semigroups, Annales de la Faculté
des Sciences de Toulouse, XXI (2012), 354-412.
[DP4] M. Dickmann, A. Petrovich, Real Semigroups, Real Spectra and Quadratic
Forms over Rings, unpublished manuscript, 2010, 256 pp.
Abstract real spectra
In Chapter 6 of
M. Marshall, Spaces of Orderings and Abstract Real Spectra, Lecture Notes Math.
1636, Springer-Verlag, Berlin, (1996).
Marshall introduces the notion of an abstract real spectrum.
These are pairs (X , G ), where X is a set and G is a subsemigroup
of 3X containing the constant functions with values 1, 0, −1
(denoted by the same symbols), and satisfying three axioms of a
topological-algebraic nature.
The main point for us is that there is a functorial duality (proved
in [DP1] above) between the category RS of real semigroups (with
obvious morphisms) and the category ARS of abstract real spectra
with morphisms defined as follows: a ARS-morphism
g : (Y , H) −→ (X , G ), is a map g : Y −→ X such that,
[ARS-mor]
For each a ∈ G the composite mapping
a ◦ g : Y −→ 3 belongs to Y .
Fans
(I) Fans in fields. Fans initially (1974) appeared as a special type
of preorders in fields:
Definition 3
(a) A (proper) preorder T on a field F is called a fan iff for all
a ∈ F × \ (−T ), we have T + Ta = T ∪ Ta. (Here,
T + Ta = {t1 + t2 a | t1 , t2 ∈ T }.)
(b) Every (total) order of a field is a fan, and the intersection of
two orders is also a fan. Fans of these types are called trivial. 2
A detailed treatment of fans in fields can be found in Ch. 5 of
T. Y. Lam, Orderings, valuations and quadratic forms, Regional Conference Series in
Mathematics 52, Amer. Math. Soc. (1983).
(II) Fans in special groups and abstract order spaces.
Later on, the notion was generalized to the abstract contexts of
reduced special groups (RSG) in
M. Dickmann, F. Miraglia, Special Groups. Boolean-Theoretic Methods in the
Theory of Quadratic Forms, Memoirs Amer. Math. Soc. 689, (2000).
and of abstract order spaces (AOS) in
M. Marshall, Spaces of Orderings and Abstract Real Spectra, Lecture Notes Math.
1636, Springer-Verlag, Berlin, (1996).
Definition 4
(a) A fan in the category RSG (henceforth a RSG-fan) is a special
group G such that 1 6= −1, whose binary representation relation
satisfies:
a ∈ DG (b, c)
iff
either b = −c or (b 6= −c and a ∈ {b, c}).
Thus, a RSG-fan is a RSG whose binary representation relation is
“smallest possible”. (A RSG-fan is necessarily a reduced special
group).
(a) A fan in the category AOS (henceforth called an AOS-fan) is
an abstract space of orders (X , G ) verifying any the following
equivalent conditions:
(1) X consists of all group homomorphisms h : G −→ { ±1} such
that h(−1) = −1.
(2) (X , G ) is an AOS and X is closed under the product of any
three of its members.
2
This amounts to saying that the set X is “biggest possible”.
(a) A fan in the category AOS (henceforth called an AOS-fan) is
an abstract space of orders (X , G ) verifying any the following
equivalent conditions:
(1) X consists of all group homomorphisms h : G −→ { ±1} such
that h(−1) = −1.
(2) (X , G ) is an AOS and X is closed under the product of any
three of its members.
2
This amounts to saying that the set X is “biggest possible”.
Of course, these notions correspond to each other under the
duality “reduced special groups/abstract order spaces” (a
particular case of the duality RS/ARS above).
Fans are building blocks in the geometrical theory of AOS, as well
as the key to many applications in real geometry; see
C. Andradas, L. Bröcker, J. Ruiz, Constructible Sets in Real Geometry, A Series of
Modern Surveys in Mathematics 33, Springer-Verlag, Berlin, Heidelberg, New York
(1996).
Fans in real semigroups and abstract real spectra.
Aim: to construct a (so far non-existing) theory of fans in the dual
categories RS and ARS and study their properties.
Fans in real semigroups and abstract real spectra.
Aim: to construct a (so far non-existing) theory of fans in the dual
categories RS and ARS and study their properties.
Method: Try to follow the model of the categories RSG and AOS.
Warning: In the case of AOS (4) fans were defined as abstract
order spaces where all characters of the underlying RSG-structure,
i.e., all group characters into {±1} sending the distinguished
element −1 to −1, are also required to preserve the additional
representation relation D. The question is: what are the
“underlying structures” in the case of real semigroups?. Answer:
ternary semigroups. Thus, we proceed as follows:
Definition 5
Given a ternary semigroup G and a non-empty set X ⊆ 3G ,
(1) (X , G ) is a fan1 iff X consists of all TS-homomorphisms from
G to 3 = {−1, 0, 1}.
(2) (X , G ) is a fan2 iff X is an ARS and is closed under the
product of any three of its members.
We shall also need the following, weaker notion to which we give a
name:
(3) (X , G ) is a q-fan (quasi-fan) iff X is closed under the product
of any three of its members and X separates points in G , i.e., for
every a, b ∈ G , a 6= b, there is h ∈ X such that h(a) 6= h(b). 2
We shall also need the following, weaker notion to which we give a
name:
(3) (X , G ) is a q-fan (quasi-fan) iff X is closed under the product
of any three of its members and X separates points in G , i.e., for
every a, b ∈ G , a 6= b, there is h ∈ X such that h(a) 6= h(b). 2
Notes. (a) We have (temporarily) chosen different names for the
two notions above, since it it is not obvious that they are
equivalent. In fact, they turn out to be equivalent under certain
(necessary) conditions, but the proof is far from trivial.
(b) In item (2) of Definition 5 we allow products of type h12 h2 ; as
opposed to the case of special groups, squaring a
TS-homomorphism does not produce a trivial map. Note also that
h3 = h, and that the product of any three TS-homomorphisms is
again a TS-homomorphism.
2
Notation. HomTS (G , 3) denotes the set of all
TS-homomorphisms from the ternary semigroup G into 3.
Fact 6
Let G be a ternary semigroup, let X ⊆ HomTS (G , 3), and assume
that (X , G ) is a q-fan. A necessary condition for (X , G ) to be an
ARS is that for all a, b ∈ G , either Z (a) ⊆ Z (b) or Z (b) ⊆ Z (a).
Here, Z (a) = { h ∈ X | h(a) = 0}.
2
This necessary condition has several characterizations:
Proposition 7
Let T be a ternary semigroup. The following conditions are
equivalent:
(1) The family {Z (a) | a ∈ T } is totally ordered under inclusion.
(2) For all a, b ∈ T , either a2 b 2 = a2 or a2 b 2 = b 2 .
(3) Every proper ideal of T is prime.
(4) The set of ideals of T is totally ordered under inclusion.
2
Fans are real semigroups and abstract real spectra
To prove these assertions, the first order of business is to work out
the explicit form of the representation relations corresponding to
the notion of “q-fan”. Recall that, given X ⊆ 3G , the relations DX
and D t are defined by the clauses
X
[R]
c ∈ DX (a, b) ⇔ ∀x ∈ X [ c(x) = 0 ∨ a(x)c(x) = 1
∨ b(x)c(x) = 1].
and
[TR]
c ∈ D t (a, b) ⇔ ∀x ∈ X [ (c(x) = 0 ∧ a(x) = b(x))
X
∨ a(x)c(x) = 1 ∨ b(x)c(x) = 1].
Fans are real semigroups and abstract real spectra
To prove these assertions, the first order of business is to work out
the explicit form of the representation relations corresponding to
the notion of “q-fan”. Recall that, given X ⊆ 3G , the relations DX
and D t are defined by the clauses
X
[R]
c ∈ DX (a, b) ⇔ ∀x ∈ X [ c(x) = 0 ∨ a(x)c(x) = 1
∨ b(x)c(x) = 1].
and
c ∈ D t (a, b) ⇔ ∀x ∈ X [ (c(x) = 0 ∧ a(x) = b(x))
X
∨ a(x)c(x) = 1 ∨ b(x)c(x) = 1].
We have:
[TR]
Theorem 8
Let G be a ternary semigroup verifying
[Z ]
∀ a, b ∈ G (a2 b 2 = a2 or a2 b 2 = b 2 ).
Let X ⊆ HomTS (G , 3) be such that (X , G ) |= q − fan.
With D = DX and D t = D t denoting the representation relations
X
defined by [R] and [TR] above, for a, b ∈ G we have:

{ a}
if Z (a) ⊂ Z (b)



{
b}
if
Z (b) ⊂ Z (a)
[D t ] D t (a, b) =
{ a, b}
if Z (a) = Z (b) and b 6= −a



{ a2 x | x ∈ G } if b = −a.
[D]
D(a, b) = a · Id(G ) ∪ b · Id(G ) ∪ {x ∈ G | xa = −xb ∧
∧ x = a2 x}. 2
With D = DX and D t = D t denoting the representation relations
X
defined by [R] and [TR] above, for a, b ∈ G we have:

{ a}
if Z (a) ⊂ Z (b)



{
b}
if
Z (b) ⊂ Z (a)
[D t ] D t (a, b) =
{ a, b}
if Z (a) = Z (b) and b 6= −a



{ a2 x | x ∈ G } if b = −a.
[D]
D(a, b) = a · Id(G ) ∪ b · Id(G ) ∪ {x ∈ G | xa = −xb ∧
∧ x = a2 x}. 2
Theorem 9
Let G be a ternary semigroup verifying condition [Z ] of Theorem
8. Then:
(1) Conditions [D] and [D t ] in Theorem 8 are interdefinable in the
following sense:
(a) Assuming that a ternary relation D on G is defined as in [D]
and the corresponding transversal representation is given by the
clause
a ∈ D t (b, c) ⇔ a ∈ D(b, c) ∧ −b ∈ D(−a, c) ∧ −c ∈ D(b, −a),
then D t verifies condition [D t ] of 8.
(b) Conversely, if D t is defined as in [D t ] and the associated
ternary representation relation D is defined by the stipulation
a ∈ D(b, c) ⇔ a ∈ D t (a2 b, a2 c),
then D verifies the equality [D] of 8.
(2) (G , D) is a real semigroup.
2
a ∈ D t (b, c) ⇔ a ∈ D(b, c) ∧ −b ∈ D(−a, c) ∧ −c ∈ D(b, −a),
then D t verifies condition [D t ] of 8.
(b) Conversely, if D t is defined as in [D t ] and the associated
ternary representation relation D is defined by the stipulation
a ∈ D(b, c) ⇔ a ∈ D t (a2 b, a2 c),
then D verifies the equality [D] of 8.
(2) (G , D) is a real semigroup.
2
The proof of these results is quite involved. However, the reward is
a large number of corollaries that to a large extent determine the
structure of the theory. Here are some of them:
Corollary 10
Let G be a TS verifying condition [Z ] of Theorem 8. Let H be a
real semigroup, and let f : G −→ H be a homomorphism of ternary
semigroups. Then, f preserves the representation relation D
defined by clause [D] of 8, and hence it is a RS-homomorphism
from (G , D) into H.
2
Corollary 11
Let G be a TS verifying condition [Z ] of Theorem 8. Then,
HomRS ((G , D), 3) = HomTS (G , 3). Hence, the ARS dual to the
real semigroup (G , D) is (HomTS (G , 3), G ).
2
Corollary 11
Let G be a TS verifying condition [Z ] of Theorem 8. Then,
HomRS ((G , D), 3) = HomTS (G , 3). Hence, the ARS dual to the
real semigroup (G , D) is (HomTS (G , 3), G ).
2
Corollary 12
Let G be a TS verifying condition [Z ] of Theorem 8 and let X ⊆
HomTS (G , 3). The following are equivalent:
(1) (X , G ) |= fan 1 (i.e., X = HomTS (G , 3) ).
(2) (X , G ) is a q-fan and verifies axiom [AX2 ] for ARSs (Marshall,
op. cit., p. 99): for every subsemigroup S of G such that
S ∪ −S = G and S ∩ −S is a prime ideal, there is h ∈ X such
that S = h−1 [ 0, 1].
(3) (X , G ) |= fan 2 .
2
Corollary 11
Let G be a TS verifying condition [Z ] of Theorem 8. Then,
HomRS ((G , D), 3) = HomTS (G , 3). Hence, the ARS dual to the
real semigroup (G , D) is (HomTS (G , 3), G ).
2
Corollary 12
Let G be a TS verifying condition [Z ] of Theorem 8 and let X ⊆
HomTS (G , 3). The following are equivalent:
(1) (X , G ) |= fan 1 (i.e., X = HomTS (G , 3) ).
(2) (X , G ) is a q-fan and verifies axiom [AX2 ] for ARSs (Marshall,
op. cit., p. 99): for every subsemigroup S of G such that
S ∪ −S = G and S ∩ −S is a prime ideal, there is h ∈ X such
that S = h−1 [ 0, 1].
(3) (X , G ) |= fan 2 .
2
Notation. Henceforth we simply write “fan” (or “ARS-fan”) for
either of the equivalent conditions fan1 or fan2 . In using the
notation (X , G ) |= fan we implicitly assume that the underlying
ternary semigroup G verifies condition [Z] in Theorem 8; this
assumption is crucial and, in fact, distinguishes fans from most
other classes of ARSs.
ternary semigroup G verifies condition [Z] in Theorem 8; this
assumption is crucial and, in fact, distinguishes fans from most
other classes of ARSs.
Corollary 13
Let (G , D) be a RS-fan. Then, the set G × = {a ∈ G | a2 = 1} of
invertible elements of G with representation induced by restriction
of D to G × , is a RSG-fan, i.e., a fan in the category of reduced
special groups.
2
ternary semigroup G verifies condition [Z] in Theorem 8; this
assumption is crucial and, in fact, distinguishes fans from most
other classes of ARSs.
Corollary 13
Let (G , D) be a RS-fan. Then, the set G × = {a ∈ G | a2 = 1} of
invertible elements of G with representation induced by restriction
of D to G × , is a RSG-fan, i.e., a fan in the category of reduced
special groups.
2
Corollary 14
Let G be a TS verifying condition [Z ] of Theorem 8 and let D be
the ternary relation on G defined by clause [D] of that Theorem.
Then,
(1) Every TS-ideal of G is a saturated ideal of the real semigroup
(G , D).
(2) A TS-subsemigroup S of G is saturated in (G , D) iff it
contains Id(G ) = {x 2 | x ∈ G } and S ∩ −S is an ideal.
2
Fans and valuation rings
Typical examples of fans amongst the RSs associated to rings are:
Theorem 15
Let A be the valuation ring of a real valuation v on a field K . The
following are equivalent:
(1) The real semigroup GA (= GA,ΣA2 ) is a fan.
(2) a) v is fully compatible with the preorder ΣA2 .
b) The residue field K has either one or two (total ) orders. 2
In fact, a more general result holds for arbitrary (ring) preorders of
A, of which Theorem 15 is a particular case:
Theorem 16
Let A be the valuation ring of a valuation v on a field K , and let
T be a (proper) preorder of A. Assume v is compatible with T .
The following are equivalent:
(1) The real semigroup GA,T is a fan.
(2) a) v is fully compatible with T .
b) The preorder T induced by T on the residue field K is
contained in at most two (total ) orders.
2
(2) a) v is fully compatible with T .
b) The preorder T induced by T on the residue field K is
contained in at most two (total ) orders.
2
Theorem 16 fits neatly with a central result from the geometric
theory of fans over fields, which we briefly recall:
Theorem (“Trivialization” of fans; Bröcker) For any fan T on a
field K , there exists a valuation v fully compatible with T such
that the pushdown T is a trivial fan of K .
2
For more details see
Lam “Orderings, Valuations . . .”,
Thms. 5.13, 12.6, and Corol. 12.7.
With this terminology, Theorem 16 characterizes the RS-fans of
type GA,T as those in which the preorder T of A is trivialized by
the (fully compatible) valuation v .
Involutions of ARS-fans
To conclude, we record two interesting properties of ARS-fans.
Definition 17
Let F be a RS-fan, let g1 , g2 ∈ XF , and fix I ∈ Spec(F ) so that
Z (g1 ), Z (g2 ) ⊆ I . We define a map ϕ g1 ,g2 : LI (F ) −→ LI (F ) as
I
follows: for h ∈ LI (F ),
ϕ g1 ,g2 (h) = h g g .
2
1 2
I
Here, for an ideal I of a RS-fan F , LI (F ) stands for the I -th level
of F , namely {h ∈ XF | Z (h) = I } (Z (h) = h−1 (0)). The levels of
a RS-fan turn out to be RSG-fans (i.e., fans in the category of
reduced special groups).
Involutions of ARS-fans
To conclude, we record two interesting properties of ARS-fans.
Definition 17
Let F be a RS-fan, let g1 , g2 ∈ XF , and fix I ∈ Spec(F ) so that
Z (g1 ), Z (g2 ) ⊆ I . We define a map ϕ g1 ,g2 : LI (F ) −→ LI (F ) as
I
follows: for h ∈ LI (F ),
ϕ g1 ,g2 (h) = h g g .
2
1 2
I
Here, for an ideal I of a RS-fan F , LI (F ) stands for the I -th level
of F , namely {h ∈ XF | Z (h) = I } (Z (h) = h−1 (0)). The levels of
a RS-fan turn out to be RSG-fans (i.e., fans in the category of
reduced special groups).
Theorem 18
With notation as in Definition 17, we have:
(a) ϕIg1 ,g2 is an AOS-automorphism of LI .
(b) ϕ g1 ,g2 is an involution: for h ∈ LI , ϕ g1 ,g2 (ϕ g1 ,g2 (h)) = h.
I
I
I
(c) For i = 1, 2, let hi be the unique
Then, ϕ g1 ,g2 (h1 ) = h2 .
I
-successor of gi in LI .
(c) For i = 1, 2, let hi be the unique
-successor of gi in LI .
g
,g
1
2
ϕ
Then,
(h1 ) = h2 .
I
In particular,
(d) If g1 , g2 , have a common
- upper bound h at some level
I ⊇ Z (g1 ), Z (g2 ), then h is a fixed point of ϕ g1 ,g2 .
I
(c) For i = 1, 2, let hi be the unique
-successor of gi in LI .
g
,g
1
2
ϕ
Then,
(h1 ) = h2 .
I
In particular,
(d) If g1 , g2 , have a common
- upper bound h at some level
I ⊇ Z (g1 ), Z (g2 ), then h is a fixed point of ϕ g1 ,g2 .
I
(e) Let J ⊆ I be in Spec(F ). Assume Z (g1 ), Z (g2 ) ⊆ J, and let
h1 ∈ LJ , h2 ∈ LI . Then,
h1
h2 ⇒ ϕ g1 ,g2 (h1 ) ϕ g1 ,g2 (h2 ).
2
J
I
(c) For i = 1, 2, let hi be the unique
-successor of gi in LI .
g
,g
1
2
ϕ
Then,
(h1 ) = h2 .
I
In particular,
(d) If g1 , g2 , have a common
- upper bound h at some level
I ⊇ Z (g1 ), Z (g2 ), then h is a fixed point of ϕ g1 ,g2 .
I
(e) Let J ⊆ I be in Spec(F ). Assume Z (g1 ), Z (g2 ) ⊆ J, and let
h1 ∈ LJ , h2 ∈ LI . Then,
h1
h2 ⇒ ϕ g1 ,g2 (h1 ) ϕ g1 ,g2 (h2 ).
2
J
I
Rather technical arguments using this result entail that the order
structure of ARS-fans is subject to rather severe constraints,
especially when it has more than one connected component .
The isomorphism theorem for finite ARS-fans
Theorem 19
Let (X1 , F1 ), (X2 , F2 ) be finite ARS-fans and let 1 , 2 denote
their respective specialization orders. If (X1 , 1 ) and (X2 , 2 ) are
order-isomorphic, then X1 and X2 are isomorphic as ARSs.
2
The isomorphism theorem for finite ARS-fans
Theorem 19
Let (X1 , F1 ), (X2 , F2 ) be finite ARS-fans and let 1 , 2 denote
their respective specialization orders. If (X1 , 1 ) and (X2 , 2 ) are
order-isomorphic, then X1 and X2 are isomorphic as ARSs.
2
In other words, for finite ARS-fans (X , F ), the specialization order
of the spectral topology in X determines completely the structure
of the fan.
The isomorphism theorem for finite ARS-fans
Theorem 19
Let (X1 , F1 ), (X2 , F2 ) be finite ARS-fans and let 1 , 2 denote
their respective specialization orders. If (X1 , 1 ) and (X2 , 2 ) are
order-isomorphic, then X1 and X2 are isomorphic as ARSs.
2
In other words, for finite ARS-fans (X , F ), the specialization order
of the spectral topology in X determines completely the structure
of the fan.
In the case of fans in the category of reduced special groups, the
analog of this result says that there is, up to isomorphism, a
unique fan in each finite cardinality. Its proof follows easily from
the existence of a structure of combinatorial geometric (matroid)
in any reduced special group; this structure was introduced and
studied in:
[D1] M. Dickmann, A Combinatorial Geometric Structure on the Space of Orders of a
Field. I, Europ. J. Combinatorics 18 (1997), 613-634.
[D2] M. Dickmann, A Combinatorial Geometric Structure on the Space of Orders of a
Field. II, in Real Algebraic Geometry and Quadratic Forms (B. Jacob, T. Y. Lam
and R. Robson, eds.), Contemporary Math. 155 (1994), AMS, 119-140.
[Li] A. Lira, Les groupes spéciaux, Ph. D. thesis, Univ. of Paris VII (1995).
[D2] M. Dickmann, A Combinatorial Geometric Structure on the Space of Orders of a
Field. II, in Real Algebraic Geometry and Quadratic Forms (B. Jacob, T. Y. Lam
and R. Robson, eds.), Contemporary Math. 155 (1994), AMS, 119-140.
[Li] A. Lira, Les groupes spéciaux, Ph. D. thesis, Univ. of Paris VII (1995).
In the present case, there is no such structure, which makes the
proof far more involved; one has to resort to a technique based on
the notion of a standard generating system.