Partial-betweenness convexity
by John Richard Ellefson
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE in Mathematics
Montana State University
© Copyright by John Richard Ellefson (1963)
Abstract:
Five postulates are given which are used to define a betweenness relation on a set. Examples are given
which show the relation is a generalization of the betweenness usually associated with real vector space
on the one hand and lattices on the other. Convex subsets are then defined on the set and shown to be
of finite character.
The extreme points of a convex subset and maximal convex sets are next defined and some of their
properties developed. The convex hull of a subset is proved to be equal to the intersection of the
maximal convex subsets. PARTIAL-BETWEENNESS CONVEXITY
by
JOHN RIGHARB ELLEFSON
A thesis submitted to the Graduate'Faculty in partial
fulfillment of the requirements:for the degree
of
MASTER OF SCIENCE
in
Mathematics
Approved:
Hea^/j, Major Department
Chairman, Examining Committee
Bean, Graduate Division
■ MONTANA STATE COLLEGE
Bozeman, Montana
June, 1963
TABLE OF CONTENTS .
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2 5
ill
ABSTRACT
Five postulates are given which are used to define a
betweenness relation on a set.
Examples are given which show
the relation is a generalization of the betweenness usually
associated with real vector space on the one hand and lattices
on the other.
Convex subsets are then defined on the set and
shown to be of finite character.
The extreme points of a convex subset and maximal convex
sets are next defined and some of their properties developed.
The convex hull of a subset is proved to be equal to the inter­
section of the maximal convex subsets.
iv
INTRODUCTION
In,order to define convexity in a set, it is necessary to
have some means of determining when a given element is between
two other elements»
Conversely, if we are able to determine
when a point is between two other points, we can define convex­
ity 0
Exactly what is meant by "between* in the above state­
ments depends on the set containing the given elements,.
If the
set is a real vector space, betweenness is usually taken as be­
ing a member of the line segment determined by two vectors«
If
the set is a lattice, betweenness can be taken as greater than
the join:and less than the meet of two elementso
It is also
possible* to postulate certain properties for bhe betweenness
relation and let these determine what betweenness is to mean*
I
The latter procedure is followed in our case*
Once we have a betweenness, we are able to define what
is'meant by a convex set*
Convex sets are, usually assumed to
have certain properties, and it is necessary to demonstrate
these properties in order to earn the full right to the name,,
I
IniSeetibns 2, 3, and 4 these properties are developed without
commentary, which makes it easier to refer to the theorems,
propositions, etc*, when they are mentioned ii| Section 5»
We will express "the element
as
b
is between
a
and
c"
abco' The union of two sets will either b$ written out or
we will use
u
placed between the sets*
Small Roman letters
will be used for elements of sets; and capital Roman letters
*™2*”
will be used for sebs^owith t’he exception of single-element Sets9
which will be designated by small Roman letters with bars over
them*
-
-
In Section I an example is given which contains numbers9
and no attempt is made to distinguish between the numbers and
vectorse
There is little danger of confusioi^, since it is ob­
vious what is to be proved and how it is to T^iq d©ne<>
I
I,
Let
E
THE POSTULATES
be a non-empty set of elements for which a be­
tweenness relation satisfying the following postulates is de­
fined »
a,
b9
and
e
are elements of
E»
abc
im-
I g a,
bp
and
c
elements of
E9
abc
and
Io
blies
cba»
±1 g n g only I g
aeb
IIIo
^
adb
ITo
Vo
b ^ x
asy
stp
are elements of
E,
abc
If
and
y
are elements of
Es
imply there exists an element
and
axc9
s
in
bsxo
a, bp Cp tp Xp
axe o bye o xty
that
d
and
Up bp Cp Xp
and
such,that
bpCp
dbco
If
byeo a / y B
E
b = c,
If a,
imply
3
Ho
and
y
are elements of
ghtgrie exists an e l g ^ ^
s
^
E
E9
such
and asb»
Some of the immediate consequences of the postulates
which will be needed in developing the properties of convex
sets defined by the betweenness relation will be derived next*
1,1
only if
If
a
and
b
are elements of
E5
aba
if and
a 53 b,
From II we see that
a - a
implies
plies
aabo
a sa b
and establishes the necessity.
baa*
which in turn im­
The.latter, combined with the given t.a b a ,
immediately from H o
implies
The sufficiency follows
lo2
If
a/’
.bj.t, x,
and
xtx
^ply
and
y
are elements of
E9
axb9
atb»
Using V 9 .we see that the hypothesis implies the existence of
ah element
s
such that
stb
and
asa,o
By I 0I 9
a = s
and
the conclusion follows*
1*3
aty
imply
Let
a = x
If
Sls b 9 t 9
and
y
are elements of
E9
ayb
and
in I «,2 and the conclusion follows immediately*
The postulates and the above properties are the only
ones we will use in the subsequent development»
It would be
interesting to investigate the implications of the postulates
more thoroughly| but the central purpose is to investigate sub­
sets having "convexity properties9" and for this goal we will
■
not need further elaboration of the postulates*
It is necessary, however, to assure ourselves that we
are not dealing with an empty theory*
To this end it seems
appropriate to construct examples satisfying the postulates*
Two examples will be given which show that the convex sets to
be defined correspond to well-known convex sets under suitable
assumptions*
A third example satisfying the postulates will
be given which indicates that these convex sets are more gen*=:;
eral than those in the other examples*
Example I *
and define
abc
Let
E
be a real or complex vector spa,ee9
as meaning there are non-negative real numbers
m
and
n
such that
b = ma + nb
and
m + n = Ie
The veri­
fication of the postulates in this example is messy but essen­
tially trivialo
We will not verify them here, but will pass
on to a more Interesting example*
Example 2 „
abe
Let
E
be a distributive lattice* and define
as meaning ordinary lattice betweenness
(6 )»
The veri­
fication of the first four postulates is again trivial*
How­
ever, the fifth one depends on constructing the element
s
and, while not difficult, is not immediately obvious*
We will,
therefore, sketch the verification of Postulate ?»
Let
ab
and
in the lattices
that
be the meet and join of
a
We wish to show that there exists an
ab ^ s ^ a + b
ap - x - a + p,
take
a + b
and
sp - t - s + p
bp - y ™ b + p,
s ** ab + at + bt*
and
and
s
xy - t - x + y*
We will
This rather obviously satisfies
To check the other relationship, notice that
t
ap - x
and
abp + etp,+ btp ^ to
(a + b + p)t ® t
bp
Now
y*
such
when we are given
s ^ a .+.,b*
because
b
We have, therefore,
ab abp -
sp 83
(s + p)t = (ab + a + b + p)t -
because
t ^ x + y ^ a + b + p *
Thus Pos­
tulate V; is verified*
Example 3 *
and define
abc
Let
E
be a filter op
as meaning
ac - b ^ a
or
the notation is the. same as in Example 2*
dual filter, or ideal, might as easily
■V-
‘
distributive lattice,
a c^b^c,
where
We note that the
have
been used*
The
6=
first three postulates are again easy to verify; and the veri­
fication of the fourth is quite similar to that of the fifth,
which we shall verify»
We wish to show thait there exists an
s - a
or
ab - s - b;
we are given that
bp y ^ p s
Let
^ t^ x
ap - x ™ a
s = ab+t,
sp - t - s
ap - x - a
and xy
Assume
and
of
or
such that
s'p - t
ap ^ x - p,
or
xy ^ t ^ y 0
and bp
~ y ^ b and
s = a b + t ^ a b + x ^ a b
abp + tp ^ xy + tp - to
s
ab ^
™ p
when
bp - y ^ b
xy ~ t
+ a-
a
- x0
and
sp =
The first chain assures us that
an element of the filter, and the second that
ing our results, we see that
sp ^ to
ab ^ ab + t = s - b
and
or
s
is
Collect­
sp ^
t ^ ab + t = s*
■ Assume
If
we let
ap - x ^ p
s'= ab, we
ter and that
and bp
see that
ab - s ^ a,
- y - p and
s
xy ^ t ^ y 0
is an element of
sp “ abp - ab ^ ab = s|
the filmand once
again Postulate 7 is verifiedo
The remaining cases are quite similar to one or the other
of the above verifications and will be assumed to avoid excess­
ive repetitiouo
The use of distributive lattices in Examples 2 and 3 is
necessary because the betweenness relation satisfies l*2o
A
theorem by Mo F t, Smiley and E» Pitcher (6 ) proves that this is
necessary and sufficient for
E
to be a distributive lattice
-7if
E
is a lattice.
the sets
E
This puts somewhat of a restriction on
in which our convex sets can be found, but not an
overly strong one.
In Example 3 we have a set which is not a
lattice even though it is a subset of a lattice, which is ade­
quate for the postulates.
It is very easy to construct examples showing the inde­
pendence of postulates on finite sets.
However, Postulate 17
can be derived from V in a real vector space, and this raises
the question of the independence of IV in a sufficiently "rich”
set
E.
The question seems to be far from trivial and is un­
answered at present.
Examples of finite sets which illustrate the independence
of Postulates IV and V are Figures I and 2 below.
Figure 3 is
a finite set satisfying all the postulates and shows that
E
need not have infinitely many points.
In the following figures an element
between
by
a
a
and
and
b
is said to be
if it lies on the line segment determined
b.
Figure I
x
Figure 2
'--Sr=;
• •
'
.
.
Gare should be taken not to confuse thd.illustrations
with the set
in general»
For instancee the intersection
of non-parallel lines need not define a unique point; there
may be other points cdmmon to the two line segments«
2»
Beflnitiono
of
E
CONVEX SETS
A line segment determined by two elements
is defined to be the set of all elements of
are between the two given elements»
If
a
elements, the line segment determined by
denoted by
and
a
E
b
and
which
are the
b
will be
Ii(a»b)«,
2<>1
Remark,
For any
empty and is eqmal to
a
and
b
in
Ep
L(a,b)
is not
L(b,a),
This follows immediately from the definition and Postulates
I and II,
2.2
longs to
Proposition,
L(a.b)»
a subset of
atb
x
b in
L( a pt)
be any element of
and
axt.
Ep
and
L( a Pt),
Applying !<,3 we get
L(apt ) ,
L ( a pt)
a similar manner we get that
if
t
be-
L( t pb)
is
By definition
ax b .
is a subset of
L(t,b)
Since
x
L(a.pb ) ,
In
is also a subset of
whence the proposition,
2.3
longs to
Proposition,
L(a.b).
is equal to
For any
a
and
b in
Ep
then the intersection of L ( a Px)
if
x
and
beL( x pb)
x.
Proof,
means
and
then the union of
Let
was arbitrary in
L ( a Pb ) p
a
L ( a pb),
Proof,
we have
For any
Let
t
be any element in the intersection.
t ; belongs to both
L (aPx)
by definition
xtb
and
axb.
combined with
axb
implies
and
L(xpb ) p
which implies
Applying 1,3 we get
axt
This
atb.
using Postulate III,
which
Since
-IQt
belongs to
L (b„x).,■ we have
when we use II0
atX n !which implies
x - t
Thus we see that any element in the intersec­
tion is equal to
x„
which is to say
x
equals the inter­
section*
2*4
if
a
Proposition*
For any
is an element of
L(dgx),
L(bnx)
a, b, c, d9
and
then the intersection of
e
and
x
in
En
is an element of
L(apd)
and
L(bsc)
is not
empty*
Proof*
b = c,
then
First consider the case
bax
and
dbxn
in
b = c
which implies
By definition we have
b
L(a»d)
section is not empty*
Similarly*
and
a 13 d
or
dba
L(bsb)
a - d*
If
by III and I*
so the inter­
implies the intersec­
tion is not empty*
Next consider the case
we have
bax
such that
and
asd
intersection of
dcx*
and
and
a ^ d*
By definition
which implies that there exists an
bse*
L(a*d)
b ^ c
Therefore*
and
L(b#c)*
s
s
is contained in the
which means that it is
not empty*
The line segments correspond to the closed intervals in
the real numbers as can be seen from properties 2*1 through 2 *4 »
In fact* since the usual definition of betweenness in the real
numbers satisfies our postulates* if
E = R*
the closed inter­
vals' will be the line segments*
Definition*
A convex subset of
E
is defined to be a
11set which contains the line segment determined by any pair of
its pointso
Rather than use the phrase,
set of
we will frequently shorten this to
E,"
vex” or speak of ’’convex
2»5
"A
is a convex sub­
nA
is con­
A”*
Remarko. E, 0,
and any single-element set of
E
are all convex*
2*6
Proposition*
For any
a
and
b
in
E,
L (a,b)
is
convex*
Proof*
t
Let
x
any element of
ment we have
y . be any elements of
L(x,y)*
axb,
which implies that
L(x,y)
and
avb.
t
and
L(a,b)|
arbitrary, this means that
2*7
Proposition*
vex sets of
Proof*
y
E
and
From the definition of a line seg­
xtv*
belongs to
is a subset of
L(a,b)
Applying 1*2 we get
L(a,b)*
and, since
L(a,b)
atb,
This implies that
x
and
y
are
is convex*
The intersection of any family of con­
is convex*
Let
J
be the family of convex sets and
any elements in the intersection*
every member of
J
and the members of
is a subset of every member of
J
Since
J
x
and
x
and
y
are in
are convex,
L(x,y)
and is, therefore, contained
in the intersection demonstrating its convexity*
Definition*
The convex hull of any given subset of
E
is defined to be the intersection of all convex sets which con­
tain the given set*
If
A
is a subset of
E,
the convex hull
*=12*”
of
A
will B.e denoted by
2o8
Ag6
Remark, 0C = 0 ,
E c = Ej
an d , if
x
belongs to
A
is a subset
Es
x c = X,
2.9
Of
Remark, If
A
is a subset of
Es
A 0.
This follows immediately from the definition,
2.10
Proposition,
If
A
is a subset
of E s A c
is
convex.
Proof,
By definition,
Ac
is the intersection of the
family of convex sets which contain
vexity of
2.11
Proposition,
If
A
JLs a subset
From 2,9 we know that
From 2,10 we see that
Ac
Proposition,
is convex if and only if
(Ac )c ,
2.13
(Ac )G
is a subset of
= A c,
(Ac )G ,
If
A
Ac
hence the proposition,
is a subset
of E,
then
A
A c = A,
Proposition 2,10 shows the necessity,
Ac
Ac
of E s
is a convex subset containing
and, ^therefore, contains
that
2,7 implies the con­
Ac,
Proof,
2.12
A,
A
being convex implies
is a subset of
A,
which shows the sufficiency,
Proposition,
If
A
and
B
are any subsets of
E,
the following statements are equivalents
a)
The convex hull of the intersection of
a subset of the intersection of
b)
The union of
Ac
and
A
and
B
is
Ac
and
Bc,
Bc
is a subset of the convex
"13hull of the union of
c)
If
Proofo
and
B
A
A
and
is a subset of
E9
let the
Set Theory then implies that
(A u B )c0
Bs
Ac
is a subset of
First we shall prove a) implies b ) »
are any subsets of
A u Bo
B»
B
Ae
Since
A
in a) stand for
is a subset of
Statement a) is symmetrical in
A
and
B 9 hence
Statement b ) »
cjo
Next we shall prove that b) implies
set of
B9
[Au B) g
Ac
Statement b) implies that
A
and
B
is contained in both
A
and
A
is a sub­
is a subset of
and that the latter is equal to
Last we prove that c) implies a)®
If
Be =
The intersection of
Bo
Statement c )9
therefore9 implies that the convex hull of the intersection of
A
and
B
is contained in both
Ae
and
Bc
and thus is in
their intersection®
2®14
Theorem®
If
A
is a convex subset of
E9
A
is
to the union of the convex hulls of all its finite sub­
sets.
Proofo
imply that
subset of
Fe
AI
Let
F
be any finite subset of
is a subset of
A®
A»
2®9 and 2®11
This is true for any finite
therefore9 it is true for their union®
If we
can now show the union of the convex hulls of all finite sub-1"
sets of
Let
A
is Convep9 we shall have completed the proof®
x
and
y
be any two elements of the union®
There
B e0
“14“
exist finite subsets
F
are contained in
and
Fe
subsets of the union of
sets of
(F u G) g
G
Ge 9
F
of
A
G9
so
according to 2 01 3 «
(F u G )c 0
such that
respectively»
and
(F u G)c p
are elements of
subset of
and
Fe
and
Since
F
and
G
y
and
G
are
x
are sub­
and
L ( x 9y)
y
is a
are finite9 their union
L ( x 9y)
is a subset of the
union of convex hulls of finite subsets of
A
whenever
x
and
are elements of this Union9 whence the property»
2ol5
Theorem*
any element of
union of all
( A u p)c
E9
A
is any subset of
then the convex hull of
where
For any
since
is convex*
If
L (a9p)
Proof *
of
and
Ge
Therefore9
which implies that
is finite; and we have shown that
y
F
x
a
a
and
a
in
p
L ( a 9p)
p
is
equals the
Ae0
is a subset of
are elements of
Therefore9 the union of all
(A u p)e *
A u p
ranges over
Ae9
E , and
(A u p )c 9
L (a9p)
which
is a subset
Inclusion the other way is demonstrated by
taking an arbitrary
x
in
(A u p)G *
set of the union of the
L( a 9p ) 9s
and each element of
shows up in an
Ac
A
is certainly a sub­
since it is a subset of
L(a9p ) *
is an element of the union since it is in each
can now show that the union of the
L C a 9P M s
Ac
Obviously
L ( a 9p ) *
p
If we
Iscoenvex9 we
will have inclusion the other way an d 9 therefore9 equality*
Let
the
x
L ( a 9p M s *
and
y
be any two elements in the union of all
There exist
a
and
b
in
Ae
such that
x
-15=
and
y
are elements of
Jj(a9p)
We want to show that
L (x9y)
t
L (x9y ) s
is any element of
and
we have axp9
bvt«
s
in
latter means that
s
is an element of L (apb)
that
Ac|
t
therefore5,
is an element of
of the union of the
Since
t
s
such that stp
is an element of
L ( s 9p)
Lfa9P ) vs
and
If
xty
Application of 7 guar­
antees an element
set of
respectively»
is a subset of the union»
by the definition of a line segment»
E
L( b Pp )9
and
as b »
The
which is
Ac0
stp
a sub­
implies
and is, therefore, an element
where
a
ranges over
Ac0
was arbitrary, we have demonstrated convexity and
proved the theoremo
3o
Definition,,
EXTREME POINTS
An element
extreme point of
A
x
for any
is in
L(a,b)
treme points of
that
e (A)
of a subset
if and only if
a
and
x = a
b
in
will be denoted by
A
or
in
E
x - b
is an
whenever
A=
The set of ex­
e (A)„
It may happen
is emptyo
301
set of
A
x
Remarko
If
A
is a subset of
Es
e (A)
is a sub­
A0
This follows immediately from the definitiono
302
If
Propositiono
A
is a subset of
Es
e(e(A)) -
e (A) o
Proofo
of
e (A)o
ment of
L ( a sb)
which is not in
there exist
and
a
x ^ a
same elements are in
e (A)o
e(e(A))
is a subset
To show inclusion the other way, let
e (A)
e ( e (A))s
in
It follows from 3»1 that
e ( e (A))o
and
and
A
b
in
x ^ bo
Since
e (A)
be any ele­
x
is not in
such that
By g d
x
is
we see that these
ands therefores that
This contradicts our assumption on
x
x
x
is not in
and establishes
the proposition0
3=3
Proposition, If
any subset of
not _in
B
subset of
is. a subset of
the set of elements in
E
Ac
and
B
is
which are
is, still convex0
Proofo
are not in
e (A)$
A
Let
Bo
A0o
x
and
y
x
and
Since
Neither
x
be any two elements in
nor
y
are in
y
is in
Ac9
B|
L ( x sy)
A
which
is a
and no element
"17"
between
sect
x
Bo
and
y
can be in
Bs
so
L (xsy)
does not inter­
If it dids there would be an extreme point between
elements which are distinct from it s which is a contradition*
Thus we have that
intersect
Bs
L(xsy)
extreme point of
Bc s
x
b
A es
is in
Proof o
A0 o
which does not
Let
and
B
Assume that x
L(asb)o
A
be a subset of
be any subset of
B
Be
is not in
x
Ac*
such that
is a subset of
e(B)j
be any
If
and let
a ^ xs
As
so
b ^ xs
Bc
x
is
which is contrary to the hypothesis*
a
and
and
x
is a subset of
Combine the results and we can conclude
e (A)s
Es
e (Be )*
be two elements of
is in
Ac
and also our proposition®
304 • Proposition*
in
is a subset of
x
is not in
We conclude the
proposition*
305
e (Ac )
Theorem*
Let
if and only if
where
B
be a subset of
x in Be
is any subset of
Proof*
E*
implies that
Then
x
x
in
Bc
of
such that
x is
is in B
By theorem
implies that there exists a finite subset
x
in
Ac *
The necessity will be proved first *
2*14»
B
A
is in
Fe
where
n
Fn
is the number of ele-
Ii
ments in
and let
Fn *
Fm
Designate the elements of
be the first m
elements of
implies that there exists an element
x
is in
L ( a sfn )*
Since
x
Fn
is in
Fn *
a .of
e (A)s
as
6 ° ° s>^n»
Theorem 2*14
F ea^
such that
it must be that
“> l S “
x = a
in
or
n - I
x
x = f p *since
is in
x = f^
or
f
is
can repeat this proc­
times until we get that
will imply that
B
We are finished if
If not, then
B0
ess
x = fn o
x
x = fg*
is in
L (f^,f2), which
In either case
x
is in
and we have proved the necessityo
To prove the sufficiency of the condition, assume that
x
is not in
and
b
e(Ac )0
If this is so, there exist elements
distinct from
ments
a
and
contains
x
b
x
such that
x
is in
L(a,b)»
constitute a t'Wo-element subset of
in its convex hullo
By assumption,
ment of the two-element set, which implies that
x =
This contradicts
bo
x
The ele­
Ag
x
which
is.an ele­
x = a
being distinct from
a
a
or
and
b
and establishes the theoremo
3 o6
Corollary o If
ment of
e(Ac ),
x
implies that
Proofo
of
Ac,
B
and
B
is in
Since
B
A
is_ a subset of
is any subset of
E,
A,
x
is an ele­
x
then
in
Bc
Bo
is a subset of
is a subset of
Ag0
A
and
A
is a subset
Apply 3=5 and the conclusion
followsO
3«?
subset of
Corollaryo If
Ac 0
3 oS
is a subset of
E,
e (Ae)
is a
A*
Proofo
in
A
A
is a subset of
3°5 implies that
Corollary=
Let
x
Ac,
is in
A
and any
x
in
e(Ac)
is
Ao
be a subset of
E0
If
A = e (Ac )s
“19"”
then for any
where
Av
equals
Proofo
also in
x
in
A
An x
is. not in the convex hull of
minus the element
in
e(A),
in
A 7c0
x»
Assume that there exists an
A 7c0 By hypothesis,
A 7 is a subset of
We conclude that
x
A
A*
x
in
A
which is
x
in
implies that
x
is
A
and, by assumption,
x
is
ds in
a contradiction, hence the conclusion*
A7
by 3=5$
but this is
4o
Befinition0
MAXIMAL CONVEX SETS
A convex subset of
E
is said to be a semi­
set if and only if its complement is also convex»
4<>1
Remark,
semiset of
4.2
E
E
is a
E,
Remark,
Definition,
of
The complement of a semiset of
E
0
and
are semisets of
E,
A semiset will be said.to separate two sets
if and only if one of the sets is a subset of the semi­
set and the other is a subset of the complement of the semiset,
4.3
Theorem,
vex subsets of
A
and
Es
(Ellis)
If
A
and
B
there exists a semiset
are disjoint con­
G
which separates
B,
Proof,
Define
M
as the set of ordered pairs
of disjoint convex subsets of
spectively,
(Gv8H 9 )
E
which contain
Induce a partial order on
if and only if
subset of
H v,
and define
n
G
M
is a subset of
A
(G8H)
and
as follows:
Gv
and
H
Consider a linearly ordered subset
N
B 8 re­
(G8H) is a
of
M8
to be the ordered pair obtained by taking the
union of all the first components of elements of
N
as the
first component and the union of all the second components of
elements of
N
maximal for
N8
as the second component.
Obviously
n
is
but we need to show that its components are
disjoint.
Assume there is an
There exist elements
a
x
and
common to the two components.
b
of
N
such that
x
is an
■=21"
element of the first component of
of the second component of
we know that
a - b
the first Case0
ponents of
tablishes
a,
n
or
bo
a fl
Since
b - a„
and
N
is an element
is linearly ordered,
To be definite, we will take
This means that
x
is an element of both com™
but these are disjoint»
as an element of
x
Ho
The contradiction es­
Since
N
was arbitrary,
we have shown that every linearly ordered subset of the par­
tially ordered set
M
M
is not empty since
has an upper bound in
Mo
(A,B)
ZornffS lemma im­
is an element»
(AffpBff)
plies that there is a maximal element
mains to- tieoshown that
Let
A ff
and
B ff0
(Aff U x ) c
and
a
and
A ff0
c
x
M0
It re­
are complementary =
B ff
and an element
a
b
common to
such that
a
respectively0
L(a,d)
are elements of
of
B ff,
L(a,d)
A ff
and
B ff0
and
L(b,e)
and
b
(Bff u x) c
common to
Using 2»14 we see that there exist a
L(d,x),
d
Bff
Assume there is an element
and
in. B ff
and
and
in
be an element in the complement of the union of
the intersection of
a
Aff
Of course,
d
are elements of
in
A ff
and
L (e,x)
We now apply 2of and conclude that
and
A ff
L(b,e)
and
b
is not emptyo
and
c
Since
are elements
are, respectively, subsets of
We, therefore, have the contradiction that an
empty set contains a non-empty subseto
It must be that at
least one of the intersections we started with is empty»
be definite, we can take the intersection of
(Aff u x ) c
To
and
“22“
Bt
as emptyo
A*p
and
Let
B*
(A^sB v )
x) c = A*»
At
is a subset of itselfo
and
mality of
(A* v
(AvsB v ) ^ (A^sB v)s
i-s a proper subset of
Therefore*
(AvsB 5) -
which contradicts the maxi-
(AvsB v)»
The contradiction tells us that
Av
and
Bv
are com­
plementary since they are disjoint and their union is the en­
tire seto
They are convex and* therefore* semisets»
Av
G
equal
4<>4
and
x
proves the theorem»
Corollary*
Let
be any element in
A
If
x
be any convex subset of
E0
exists a semiset separating
Proof»
Letting
If
x
x
is, not in
and
is not in
As
As
E
there
A»
x
and
A
are disjoint
convex subsets to which we can apply 4»3o
4*5
Corollary *
an extreme point of
A
only at
Av
A9
A
is, any subset of
E
and
x
is
then there exists a semiset meeting
X0
Proof*
3*39
If
Let
Av
equal
is convex and
x
A
and
minus the extreme point„
Av
By
are disjoint convex sets*
Apply 4<>3 and the corollary follows*
4*6
Theorem*
If
A
is any subset of
Es
the intersection of all semisets which: ,contain
Proof*
If
also a subset of
A
G*
is a subset of any semiset
From this we conclude.that
Ac
equals
A*
Cs
Ac
Ae
is
is a
subset of the intersection of all semisets which contain
A*
“23*=
To show inclusion the other way, we consider an
By 4=3# there is a semiset
A t*
is a subset of
that
A
taining
G0
G
A
such that
is a subset of
is also a subset of
A
G0
x
is not in
A „
G
A0o
and
which means
x
and can thereby con­
is not ah element of the intersection of all
semisets containing
4=7
not in
We have found a semiset con­
as a subset which excludes
clude that
x
x
A
Theorem*
as a subset c
Let
A
be any convex subset of
E0
A
is equal to the convex hull of its extreme points if and only
if every semiset of
O M S M B H aa ad a a*
Mo»aMM*MnaMM*HHMa»
Proofo
E
so ^ aao
which intersects
M so asM ^sa^M M
ment of
y
C0
a subset of
that
A
intersects
The necessity will be proved first»
there exists an element
such that
(M n o ^ M M a a i^ o o M o ia M a a n a a M M
is in
Since
C
y
G
of
and
e (A)
A
e(A)
G,
G
in
(e(A))c
E
is also
From this we can conclude
Thereforep the necessity is proved*
The sufficiency is proved by assuming that
If this is s o s we can find an
(e(A))G 0
Assume
is a subset of the comple­
is a subset of
by 2<,13 and 2*11*
A / ( e ( A ))c 0
and a semiset
e(A)o
x
in
A
A ^ (e(4))c »
which is not in
Apply 4«4 and we are Assured of the existence of a
semiset separating
ciency is proved„
x
and
(e(A))e 0
Therefore, the suffi­
5o
COMMENTARY
Special attention is called to 2*14, for it shows that
the convex sets are of finite character, and to 4#3 in which
the existence of separating semisets is proved*
These are in
agreement with the results on a real vector space*
One should
recall, however, that the betweenness relation is valid for a
finite set of elements*
Many of the properties of convex sets
in a real vector space are found for the convex sets which we
have defined and justify the use of the name*
The possibility that the set
E
has extreme points pre­
cludes defining a subset of the semisets having only one ex­
treme point*
By insisting the set of extreme points of
E
be
empty, we could define an analogy to convex cones and take the
semisets which are cones as a minimal intersection basis*
The betweenness relation introduced in Example 3 has the
advantage of having the elements which define a line segment
as the extreme points of the segment*
This is not necessarily
true in the betweenness relation of Example 2*
If the meet
and join of two elements are distinct from the elements, the
line segment determined by the two elements will have no e x ­
treme points* . Since the line segments correspond to closed
\
intervals in a real vector space, it makes the correspondence
a closer one if all line segments have extreme points*
LITERATURE CONSULTED
Io
L 0 Danzerp B 6 Grunbaump Vo Kleep Hellygs theorem and its
relatives. Unpublished Monograph*
20
Jo Wo Ellisp A general set-separation theorem* Duke Mathe J e
volo 19 (1952) ppo 417-21o
—
30
So Po Franklinp Some results on order convexity„ Amere Ma t h e
Monthly vol® 69 No* 5 (1963) PP® 357-9«
4o
Wo Grevep Partial betweenness groups„ Math* Z 0 vol* 7&
(1962) ppo 305-18.
5o
Po Co Hammer, Maximal convex sets, Duke Math* J 0 vol* 22
(1955) PPo 103-6o
6
Everett Pitcher, Mo F e Smiley, Transitivities of betweenvs n
Arvi
4- V\
Q z\zt
TrzsT
I
. ____
- , . T r IHdTUPRSTTY LIBRARIES
3 1 7 6 2 IbOI3656 1
NS 78
EI54
cop .S
[Ellefson, John R .
PartiaI-betweenness convexity.
NAMK ANP AOPWKSa
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NVWr
aSaiWO ^
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