Generating geometry axioms from poset axioms

0.1 MB·

by  Wolfram Retter

GENERATING GEOMETRY AXIOMS FROM POSET AXIOMS
WOLFRAMRETTER
4 ABSTRACT. Twoaxiomsofordergeoemtryaretheposetaxiomsoftransitivityandantisymmetry
1 of the relation’is in frontof’when lookingfroma point. From these axioms, by lookingfrom
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anintervalinsteadofapoint,furtherwell-knownaxiomsofordergeometryaregeneratedinthe
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following sense: Transitivity when looking from an interval is equivalent to [4, §10, Assioma

n
XIII].Assumingthisaxiom,antisymmetrywhenlookingfromanintervalisequivalentto[3,§1,
a
VIII. Grundsatz]. Further equivalences, with some of the implications well-known, are proved
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alongtheway.
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C CONTENTS
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1. Introduction 1
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a 2. Interval-TransitivityCriterion 4
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3. Interval-AntisymmetryCrtierion 8
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4. Conclusion 11

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References 11
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. 1. INTRODUCTION
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0 Let X be a vectorspace overa totallyordered field K, for exampleK = R and X = Rn for an
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1 n ∈ Z≥1.Thevector intervalrelationonX istheternary relationh·, ·, ·idefined by
:
v hx, y, zi :⇔ Thereisa λ ∈ K such that0 ≤ λ ≤ 1and y = x+λ(z −x) ,
i
X
X togetherwiththisrelationsatisfies thefollowingconditions:
r
a
◦ Fora ∈ X,thebinaryrelationsh·, ·, ai and ha, ·, ·iare reflexiveonX .
◦ Fora ∈ X,thebinaryrelation h·, a, ·iis symmetric.
◦ Forx, y ∈ X,hx, y, xi impliesy = x.
An interval space is a pair consisting of a set X and a ternary relation h·, ·, ·i on X such that
theseconditionsaresatisfied. Thus,avectorspaceX overatotallyorderedfieldK togetherwith
its vector interval relation is an interval space. The concept of an interval space has been taken
from [9,chapterI, 3.1].
January 12, 2014
Date: .
2010MathematicsSubjectClassification. 51D20,51G05,52A01.
Keywordsandphrases. intervalspace,ordergeometry,orderedgeometry,poset,antimatroid.
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2 WOLFRAMRETTER
An interval space (X, h·, ·, ·i) is also simply denoted by X when it is clear from the context
whethertheintervalspaceoronlytheset ismeant.
An interval space X is called point-transitiveiff for each a ∈ X , the binary relation ha, ·, ·i
is transitive, i.e. for all x, y, z ∈ X , (ha, x, yi and ha, y, zi) =⇒ ha, x, zi . A vector space
with its vector interval relation is point-transitive and satisfies the equivalent conditions of the
followingtheorem. Condition(1)isobtainedfromthepoint-transitivityconditionthatha, ·, ·iis
transitivebyreplacing thepointabyan interval. Condition(2)istheintervalrelationversionof
thestrictintervalrelationcondition[4, §10,AssiomaXIII].
•c
♦♦♦♦♦♦♦x♦♦•♦♦♦♦•
• • •
a b
Thedefinitionsofthe intervalspaceconcepts and notationsfollowimmediatleyafterwards. The
proof is givenbelow. For counter-examples and more examples and history of the concepts, see
[6, sections 1.4, 1.5] and the references given there. For a set X, P (X) denotes the power set
ofX,i.e. thesetofall subsetsofX .
Theorem. 2.3 (interval-transitivity criterion) Let X be an interval space. Then the following
conditionsareequivalent:
(1) X isinterval-transitive.
(2) Foralla, b, c ∈ X ,[{a}, [b, c]] ⊆ [[a, b], {c}] .
(3) Foralla, b, c ∈ X ,[{a}, [b, c]] = [[a, b], {c}] .
(4) P (X) withthebinaryoperation[·, ·]isa semigroup.
(5) P (X) withthebinaryoperation[·, ·]isa commutativesemigroup.
(6) X isinterval-convex,andforeachconvexsetA,thebinaryrelationhA, ·, ·iistransitive.
(7) Forallconvex setsA, B, [A, B]isconvex.
(8) Foralla, b, c ∈ X ,[[a, b], {c}]is convex.
(9) Foralla, b, c ∈ X ,co({a, b, c}) = [[a, b], {c}] .
Let X bean intervalspace.
ForA ⊆ X and b, c ∈ X ,
hA, b, ci :⇐⇒ There isan a ∈ Asuchthat ha, b, ci .
ForA, C ⊆ X andb ∈ X,
hA, b, Ci :⇐⇒ Therearea ∈ A, c ∈ C suchthat ha, b, ci .
Fora, c ∈ X,theintervalbetween aand c istheset
[a, c] := ha, ·, ci
= {x ∈ X| ha, x, ci} .
ForA, C ⊆ X,theintervalbetween Aand C istheset
GENERATINGGEOMETRYAXIOMSFROMPOSETAXIOMS 3
[A, C] := hA, ·, Ci
= {x ∈ X| hA, x, Ci} .
A subset C of X is called convex iff [C, C] ⊆ C, i.e. for all x, y, z ∈ X , if hx, y, zi and
x, z ∈ C,theny ∈ C.
ForA ⊆ X ,theconvex closureorconvex hullofAinX istheset
co(A) := \{B ⊆ X|B ⊇ Aand B isconvex.} .
It isthesmallestconvexsetin X containgA.
X is called interval-transitive iff for for all a, b ∈ X , the binary relation h[a, b], ·, ·i is
transitive. Each interval-transitiveintervalspaceis point-transitive.
X is called interval-convex iff for all a, b ∈ X , [a, b] is convex. The concept of an interval-
convex interval space generalizes the concept of an interval monotone graph in [2, 1.1.6]. The
term ’interval-convex’hasbeen introducedin [6,section 1.5].
An interval space X is called point-antisymmetric iff for each a ∈ X, the binary relation
ha, ·, ·iisantisymmetric,i.e. forallx, y ∈ X \[a, b] ,(ha, x, yi and ha, y, xi) =⇒ x = y.A
vector space with its vector interval relation is point-antisymmetric. It is also interval-transitive
and satisfies the equivalent conditions of the following theorem. Condition (1) is obtained from
the point-antisymmetry condition that ha, ·, ·i is antisymmetric by replacing the point a by an
interval. Condition (2) is the interval relation version of the strict interval relation condition [3,
§1, VIII. Grundsatz]. The definitions of the interval space and closure space concepts follow
immediatley afterwards. The proof is given below. For counter-examples and more examples
and history of the interval space concepts, see [6, sections 1.4, 1.5] and the references given
there.
Theorem. 3.5 (interval-antisymmetry criterion) Let X be an interval-transitive interval space.
Then thefollowingconditionsareequivalent:
(1) X isinterval-antisymmetric.
(2) X isstiff.
(3) Foreach convexset A,thebinaryrelationhA, ·, ·iis antisymmetriconX \A.
(4) Thepairconsistingof X andthesetof convexsetsis anantiexchangespace.
(5) Thepairconsistingof X andthesetof convexsetsis anantimatroid.
Let X bean intervalspace.
X is called interval-antisymmetric iff for all a, b ∈ X the binary relation h[a, b], ·, ·i is
antisymmetriconX\[a, b] .Ingeneral,therewillbenocahncethatthisrelationisantisymmetric
on thewholeofX .
X is called stiff iff for all a, b, c, d ∈ X, (ha, b, ci and b 6= c and hb, c, di) =⇒ ha, b, di .
Asnotedabove,thisconditionistheintervalrelationversionofthestrictintervalrelationcondi-
tion[3, §1,VIII. Grundsatz]. In [6, section3.2]astiffintervalspacehas been called one-way.
• • • •
a b c d
4 WOLFRAMRETTER
Let X be a set. A closure system or Moore family on X is a set C of subsets of X such that
X ∈ C and foreach non-emptyD ⊆ C, D ∈ C.
T
A closurespaceis a pairconsistingof aset X and a closuresystemC on X .A set A ⊆ X is
called closed iffA ∈ C. When (X, O) is a topologicalspace, then thepair consistingof X and
the set of closed sets in (X, O) is a closure space. When (X, h·, ·, ·i) is an interval space, then
the pair consisting of X and the set of convex sets is a closure space. The concept of a closure
space as defined here is slighly more general than in [8, chapter I, 1.2], where it is required that
∅ ∈ C and aclosuresystemiscalled aprotopology.
Aclosurespace(X, C)isalsosimplydenotedbyX whenitisclearfromthecontextwhether
theclosurespaceoronlytheset ismeant.
Let(X, C) beaclosurespace.
ForA ⊆ X ,theclosureofAistheset
cl(A) := \{B ⊆ X|B ⊇ Aand B ∈ C}
ItisthesmallestclosedsupersetofA.WhenX isanintervalspaceandC isthesystemofconvex
setsin X,then forA ⊆ X,theclosureofAis theconvexclosureofA.
For A ⊆ X , the entailment relation of C relative to A or A-entailment relation is the binary
relation⊢A on X defined by
x ⊢A y :⇔ y ∈ cl(A∪{x}) .
(X, C)is called an antiexchange space iff for each closed A ⊆ X, one and therefore all of the
followingconditionshold,whichare equivalentby[6, Proposition3.1.1]:
◦ Therelation⊢A isantisymmetricon X \A.
◦ Therestriction⊢A |(X \A) is apartialorder onX \A.
(X, C) is called algebraicorcombinatorialifffor each chain D ⊆ C, D ∈ C. [8, chapter I,
S
1.3] states theequivalenceof this definition with otherwell-knowndefinitions. A combinatorial
(i.e. algebraic) closure space is also just called a combinatorial space. When X is an interval
spaceand C isthesystemofconvexsetsin X,then (X, C) isa combinatorialclosurespace.
An antimatroid (anti-matroid) or Dilworth space is a combinatorial exchange space with ∅
closed. Thisconcept has been taken from[8, chapterI, 2.24].
2. INTERVAL-TRANSITIVITY CRITERION
Part (1) of the following proposition is cited from [5, Theorem 2.3]. Parts (2) and (3) are cited
from [5,Theorem 2.1].
Proposition 2.1. (set interval operator) Let X be an interval space. The binary operation [·, ·]
on P (X) hasthefollowingproperties:
(1) [·, ·]iscommutative,i.e. forA, B ⊆ X,[A, B] = [B, A] .
(2) For C ⊆ X , the unary operation [·, C] is increasing, i.e. for A, B ⊆ X, A ⊆ B =⇒
[A, C] ⊆ [B, C] .
(3) For C ⊆ X , the unary operation [C, ·] is increasing, i.e. for A, B ⊆ X, A ⊆ B =⇒
[C, A] ⊆ [C, B] .
GENERATINGGEOMETRYAXIOMSFROMPOSETAXIOMS 5
(4) The binary operation [·, ·] is increasing, i.e. for A, B, C, D ⊆ X, A ⊆ B and C ⊆
D =⇒ [A, C] ⊆ [B, D] .
Proof.
(1) [5, Theorem2.3]
(2) [5, Theorem2.1]
(3) [5, Theorem2.1]
(4) followsfrom (3)and (4)
(cid:3)
Let X be an intervalspace and a, b ∈ X .If the binary relation h[a, b], ·, ·i is transitive,then
forx, y, c ∈ X,
h[a, b], x, yi and h[a, b], y, ci =⇒ h[a, b], x, ci .
Substitutinga ∈ [a, b] andb ∈ [a, b] ,
ha, x, yi and hb, y, ci =⇒ h[a, b], x, ci ,
i.e.
ha, x, yi and y ∈ [b, c] =⇒ h[a, b], x, ci .
Consequently,forx, c ∈ X,
ha, x, [b, c]i =⇒ h[a, b], x, ci ,
i.e. forc ∈ X ,
[{a}, [b, c]] ⊆[[a, b], {c}] .
The last condition says: For x ∈ X, if x is between a and [b, c] , then x is also between [a, b]
and c :
•c
♦♦♦♦♦♦♦x♦♦•♦♦♦♦•
• • •
a b
Summarizing,
Proposition 2.2. (interval spaces transitive from a base-interval) Let X be an interval space
and a, b ∈ X. If the binary relation h[a, b], ·, ·i is transitive, then for c ∈ X, [a, [b, c]] ⊆
[[a, b], c] .
Condition(2)inthefollowingtheoremistheintervalrelationversionofthestrictintervalrelation
condition[4, §10, AssiomaXIII]. In [8, chapter I, 4.9] it has been called thePeano Property. In
[6, section 1.5], its equivalence with conditions (8) and (9) has been stated, and accordingly, X
has beencalled triangle-convexiffoneand thereforeeach ofthesethreeequivalentconditionsis
satisfied. The implication (3) ⇒ (4) has been proved in [5, Theorem 2.4]. In [8, chapter I, 4.10
6 WOLFRAMRETTER
(2)] it has been shown that (2) implies interval-convexity in (6). In [5, Theorem 4.45 (a)] it has
been demonstrated that (3) implies the strict interval relation version of transitivity in (6). The
implication(2)⇒(7)has been provedin[5, Theorem 2.12].
Theorem 2.3. (interval-transitivity criterion) Let X be an interval space. Then the following
conditionsareequivalent:
(1) X isinterval-transitive.
(2) Foralla, b, c ∈ X ,[{a}, [b, c]] ⊆ [[a, b], {c}] .
(3) Foralla, b, c ∈ X ,[{a}, [b, c]] = [[a, b], {c}] .
(4) P (X) withthebinaryoperation[·, ·]isa semigroup.
(5) P (X) withthebinaryoperation[·, ·]isa commutativesemigroup.
(6) X isinterval-convex,andforeachconvexsetA,thebinaryrelationhA, ·, ·iistransitive.
(7) Forallconvex setsA, B, [A, B]isconvex.
(8) Foralla, b, c ∈ X ,[[a, b], {c}]is convex.
(9) Foralla, b, c ∈ X ,co({a, b, c}) = [[a, b], {c}] .
Proof. Step 1. (1)⇒ (2)followsby 2.2(intervalspaces transitivefromabase-interval).
Step2. (2)⇒(3). Fora, b, c ∈ X itremainstobeprovedthat= [[a, b], c] ⊆ [a, [b, c]] .The
assumption(2)impliesby 2.1(1)(set intervaloperator):
[[a, b], {c}]
=[{c}, [b, a]]
⊆[[c, b], {a}]
=[{a}, [b, c]] .
Step 3. (3)⇒ (4). [5, Theorem2.4]
Step 4. (4)⇒ (5)followsby 2.1(1)(set intervaloperator).
Step 5. (5)⇒ (6).
Step 5.1. Proof that X is interval-convex, i.e. for a, b ∈ X, [[a, b], [a, b]] ⊆ [a, b] . The
assumption(5)impliesby generalized associativityand commutativity:
[[a, b], [a, b]]
=[[{a}, {b}], [{a}, {b}]]
=[[{a}, {a}], [{b}, {b}]]
=[{a}, {b}]
=[a, b] .
Step 5.2. Proof that for each convex set A, the binary relation hA, ·, ·i is transitive, i.e. for
x, y, z ∈ X, hA, x, yi and hA, y, zi implies hA, x, zi , i.e. for y, z ∈ X, y ∈ [A, {z}]
implies[A, {y}] ⊆ [A, {z}] .Theassumptionthat Aisconvexsays:
[A, A] ⊆ A. (2.1)
GENERATINGGEOMETRYAXIOMSFROMPOSETAXIOMS 7
From theassumptionsy ∈ [A, {z}] ,i.e. {y} ⊆ [A, {z}] ,and (5)and (2.1)it followsby2.1(2)
and (3)(set intervaloperator):
[A, {y}]
⊆[A, [A, {z}]]
=[[A, A], {z}]
⊆[A, {z}] .
Step 6. (6)⇒ (1). Theassumptionthat X is interval-convexentails:
[a, b] isconvex. (2.2)
From(2.2)andtheassumptionthatforeachconvexsetAthebinaryrelationhA, ·, ·iistransitive
itfollowsthath[a, b], ·, ·iistransitive.
Step 7. (5)⇒ (7). Theassumptionthat A, B are convexsays:
[A, A] ⊆ A, (2.3)
[B, B] ⊆ B. (2.4)
The assumption (5), (2.3) and (2.4) imply by by generalized associativity and commutativity:
and 2.1(4)(set intervaloperator):
[[A, B], [A, B]]
=[[A, A], [B, B]]
⊆[A, B] .
Step 8. (7)⇒ (8).
{a}, {b}, {c} are convex. (2.5)
From (2.5)andtheassumption(7)itfollowsthat [{a}, {b}]isconvex,i.e.
[a, b] isconvex. (2.6)
From (2.6)andtheassumption(7)implythat [[a, b], c]isconvex.
Step 9. (8)⇒(9). Theassumption(8) impliesthatit suffices toprovethatfor C a convexset,
C ⊇ {a, b, c} iffC ⊇ [[a, b], c] .From theassumptionthatC isconvexitfollows:
C ⊇ {a, b, c}
⇐⇒(C ⊇ {a, b} and C ⊇ {c})
⇐⇒(C ⊇ [a, b] andC ⊇ {c})
⇐⇒C ⊇ [[a, b], c]
8 WOLFRAMRETTER
Step 10. (9)⇒ (3). Theassumption(9)impliesby2.1(1)(set intervaloperator):
[{a}, [b, c]]
=[[b, c], {a}]
=co({b, c, a})
=co({a, b, c})
=[[a, b], {c}] .
(cid:3)
3. INTERVAL-ANTISYMMETRY CRTIERION
Let X be point-transitive interval space and a, d ∈ X . If the binary relation h[a, d], ·, ·i is
antisymmetriconX \[a, d] ,thenfor b, c ∈ X,
(c, b ∈/ [a, d] and h[a, d], b, ci and h[a, d], c, bi) =⇒ b = c.
Substitutinga ∈ [a, d]inthesecondand and d ∈ [a, d]inthethirdcondition,
(c, b ∈/ [a, d] and ha, b, ci and hd, c, bi) =⇒ b = c.
Equivalently,
(ha, b, ci and hd, c, bi and b 6= c) =⇒ (c ∈ [a, d] orb ∈ [a, d]) .
Rewriting,
(ha, b, ci and b 6= c and hb, c, di) =⇒ (ha, c, di or ha, b, di) .
The condition ha, b, ci on the left, the first possibility ha, c, di on the right and the assumption
thatX ispoint-transitiveimplythesecond possibilityhb, c, dion theright. Consequently,
(ha, b, ci and b 6= c and hb, c, di) =⇒ ha, b, di .
Summarizing,
Proposition3.1. (intervalspacesantisymmetricfromabase-interval)LetX beapoint-transitive
intervalspaceanda, d ∈ X.Ifthebinaryrelationh[a, d], ·, ·iisantisymmetriconX \[a, d] ,
thenfor b, c ∈ X ,(ha, b, ci andb 6= c and hb, c, di) =⇒ ha, b, di .
Thefollowingpropositionis similarto[1, chapterII, proposition10].
Proposition 3.2. (stiff interval spaces) Let X be a stiff interval space. Then for each convex set
A,thebinaryrelationhA, ·, ·i isantisymmetricon X \A.
Proof. It is to be proved that b, c ∈ X \A, hA, b, ci and hA, c, bi implies b = c. The assump-
tionshA, b, ciand hA, c, bi saythat thereare
a, d ∈ A (3.1)
such that
ha, b, ci (3.2)
GENERATINGGEOMETRYAXIOMSFROMPOSETAXIOMS 9
and hd, c, bi ,i.e.
hb, c, di . (3.3)
Seeking acontradiction,suppose
b 6= c. (3.4)
(3.2), (3.4), (3.3)and theassumption(2)imply:
ha, b, di . (3.5)
From (3.1), (3.5) and the assumption that A is convex it follows that b ∈ A, contradicting the
assumptionthatb ∈ X \A. (cid:3)
Proposition 3.3. (interval-transitve interval spaces) Let X be an interval-transitive interval
space and A a convex set. Then the relative entailment relation ⊢A is the reverse relation of
thebinaryrelationhA, ·, ·i .
Proof. It is tobeprovedthat forb, c ∈ X , c ⊢A biffhA, b, ci .
{c} isconvex. (3.6)
From (3.6) and the assumptions that A is convex and X is interval-transitive it follows by 2.3
(interval-transitivity criterion) that [A, {c}] is convex. Therefore, co(A∪{c}) = [A, {c}] .
Consequently,thefollowingequivalenceshold:
c ⊢A b
⇐⇒b ∈ co(A∪{c})
⇐⇒b ∈ [A, {c}]
⇐⇒hA, b, ci .
(cid:3)
10 WOLFRAMRETTER
Thefollowingpropositionisaparticularcaseofamoregeneralprincipleforrelationalstructures.
Proposition 3.4. (interval spaces are combinatorial spaces) Let X be an interval space. Then
theclosurespaceconsistingofX andtheset ofconvex setsiscombinatorial.
Proof. For a chain D of convex sets is to be proved that D is convex, i.e. for a, b, c ∈ X, if
S
a, c ∈ D and ha, b, ci , then b ∈ D. The assumption that a, c ∈ D says that there are
S S S
A, C ∈ D suchthat
a ∈ A, (3.7)
c ∈ C. (3.8)
FromtheassumptionsthatDisachainandA, C ∈ D itfollowsthatA ⊆ C orC ⊆ A.Suppose
withoutlossofgeneralitythat
A ⊆ C. (3.9)
(3.7)and (3.9)imply
a ∈ C. (3.10)
From (3.10), (3.8)and theassumptionsthatha, b, ci and C isconvexitfollows:
b ∈ C. (3.11)
(3.11)andtheassumptionthatC ∈ D implythatb ∈ D. (cid:3)
S
Theorem 3.5. (interval-antisymmetry criterion) Let X be an interval-transitive interval space.
Then thefollowingconditionsareequivalent:
(1) X isinterval-antisymmetric.
(2) X isstiff.
(3) Foreach convexset A,thebinaryrelationhA, ·, ·iis antisymmetriconX \A.
(4) Thepairconsistingof X andthesetof convexsetsis anantiexchangespace.
(5) Thepairconsistingof X andthesetof convexsetsis anantimatroid.
Proof. Step 1. (1)⇒ (2). FromtheassumptionthatX isinterval-transitiveitfollows:
X ispoint-transitive. (3.12)
From (3.12) and the assumption (1) it follows 3.1 (interval spaces antisymmetric from a base-
interval)thatX isstiff.
Step 2. (2)⇒ (3)followsfrom 3.2(stiffintervalspaces).
Step 3. (3) ⇒ (1). For a, d ∈ X it is to be proved that the binary relation h[a, d], ·, ·i is
antisymmetriconX \[a, d] .From theassumptionthatX isinterval-transitiveitfollowsby2.3
(interval-transitivitycriterion)thatX isinterval-convex. In particular,
[a, d] is convex. (3.13)
(3.13) and the assumption (3) imply that the binary relation h[a, d], ·, ·i is antisymmetric on
X \[a, d] .
Step 4. (3) ⇔ (4). It is to be proved iff that for each convex set A, the binary relation
hA, ·, ·iisantisymmetriconX\A.foreachconvexsetA,therelativeentailmentrelation⊢A is