AN ABSTRACT OF THE THESIS OF
DENNIS IRL GOSLIN
for the
in
(Degree)
(Name)
Date thesis is presented
Title
M. S.
Mathematics
(Major)
August 13, 1965
THE MATHEMATICAL STRUCTURE OF RECTANGULAR
ARRANGEMENTS
Abstract approved
Redacted for Privacy
(Major Professor)
The author studies the class of rectangular arrangements in
terms
of two
binary relations on the objects of the arrangement.
He shows how a univalent
matrix determines a unique rectangular
arrangement, and how each rectangular arrangement is associated
with one, two, or four distinct matrices, according to the number
of
corner elements of the arrangement. In this manner it is shown
that the essence of rectangular arrangements can be captured in
mathematical terms.
THE MATHEMATICAL STRUCTURE OF
RECTANGULAR ARRANGEMENTS
by
DENNIS IRL GOSLIN
A THESIS
submitted to
OREGON STATE UNIVERSITY
partial fulfillment of
the requirements for the
in
degree of
MASTER OF SCIENCE
June 1966
APPROVED:
Redacted for Privacy
Professor
of
Mathematics
In Charge of Major
Redacted for Privacy
Chairman of Department of Mathematics
Redacted
for Privacy
Dean of Graduate School
Date thesis is presented
Typed by Carol Baker
August 13, 1965
ACKNOWLEDGEMENT
The author would like to thank Dr. D.S. Carter for suggesting
this problem and for his advise and encouragement. He would also
like to thank Professor J. Davis for critically reading this paper.
THE MATHEMATICAL STRUCTURE OF
RECTANGULAR ARRANGEMENTS
CHAPTER I
INTRODUCTION
Mathematics is the study of objects of thought and relations on
the objects. In this paper we will discuss a class of "simple mathe-
matical structures" corresponding to rectangular arrangements
objects. We will abstract the study
lar arrangements to the study
of
class
of the
of
relations on sets
of
all such rectanguof
objects, inde-
pendent of the nature of the objects. In this manner we will capture,
in mathematical
terms, the essence
of the
rectangular arrangement.
The following are examples of rectangular arrangements:
Example
1.
The arrangement of the elements in a matrix.
The rows and columns are specified by the definition of a matrix
(see Definition 1.8 below).
Example
2.
The arrangement of trees in rows and columns as in
an orchard.
Example 3.
The arrangement of people into rows and columns
as in a football stadium.
Example 4.
The arrangement of the Gaussian integers
where m and n are integers such that
0 <
m
<
k,
0 < n <
m+ ni
,Q
.
2
As these examples show, the essential nature of a rectangular
arrangement is quite independent
particular objects which
The essential thing is a set of objects
make up the arrangement.
of any kind,
of the
arrayed in rows and columns. To explain
how this
abstract notion can be captured in mathematical terms,
first present some basic definitions leading
we must
up to the idea of a simple
mathematical structure.
Definition
1. 1.
Let
n
be a positive integer,
set. A sequence of length n in
of
X
(xl, x2,
notes the value of the function at the integer
,
into the set
X.
Such
where
xi
de-
xn)
for
i,
The set of all such n -ary sequences is denoted by
Definition
1. 2.
An n -ary relation on
Let
X
n
be a
X
is a function, defined on the set
positive integers less than or equal to n,
sequences are usually denoted by
and
i
Xn.
A
1, 2
,
,
1
X
be a set.
-ary relation is
called a unary relation, a 2 -ary relation is called a binary relation,
and a
3
-ary relation is called a ternary relation.
The following are examples of relations on the set of all
tri-
angles.
Example
1.
Unary relation: The set of all
is an equilateral triangle.
n.
Xn.
be a positive integer and
is a subset of
=
x
such that
x
3
Example
Binary relation: The set
2.
and
x
of
pairs
such that
(x, y)
are congruent triangles.
y
Example 3. Ternary relation: The set of triples
that the area of
the area of
(x, y, z)
plus the area of
x
such
is equal to
y
z.
The reader should be familiar with the following types of binary
relations:
Definition 1.3.
binary relation
A
ric if for all elements
Definition
if for
1.
4.
all elements
Definition
A
a, b
of
A
binary relation
1. 5.
A
then
Definition
set
is symmet-
then
(b, a)E A.
X
is reflexive
a, b,
c
of
on a set
A
if
X,
(a, b)
X
and
is
transi-
(b, c)
be-
(a, c)E A.
A
1. 6.
alence relation if
A on a
X
(a, a)E A.
a
X,
(a, b)EA
if
X,
binary relation
tive if for all elements
long to
of
on a set
A
A
binary relation
A
on a set
X
is an equiv-
is reflexive, symmetric and transitive.
In introducing the notion of a simple
mathematical structure, we
shall first present the abstract definition, followed by two familiar
examples from algebra.
4
Definition 1.7.
Let
be a set of objects and
X
be a finite set of relations on
X
is the set
relations on
(X,A1,A2,
X
together with the finite set
{A
A2,
,An} of
The mathematical structure is denoted by
X.
.
,A1}
simple mathematical structure on
A
X.
{A1,A2,
,An).
The relations
are called the distinguished
Ai
relations of the structure.
Example
where
X
1.
A group is a
mathematical structure
is the set of elements of the group,
E
(X, E, M, I)
is the unary
re-
lation consisting of the identity e:
E
M
{e
},
is a ternary relation, the group operation (multiplication):
M
and
=
I
=
{(x,y,z):z
=
xy },
is a binary relation, the group inverse:
I
Example
structure
2.
(X, U, S)
=
{(x,y):xy
=
e }.
The natural numbers form a simple mathematical
where
X
is the set of natural numbers,
is a unary relation consisting of the element
U={1},
1:
U
5
and
is a binary relation, the successor:
S
S
=
successor
{(x, y): y is the
of
x}
.
Finally, we define the notion of a matrix in a set
Definition
integers.
Let
tive integers
1. 8.
Let
mX n
(i, j)
X
be a set,
m, n
be positive
denote the set of all ordered pairs of posi-
such that
i <
is a function defined on the set
X
and
X.
m,
j <
mX n
n.
An
mX n matrix in
into the set
X.
Such
matrices are often represented by rectangular arrays as in Figure
1. 1,
xln
x11 x12
x21 x22
.
xZn
'
.
xmlxm2'
xmn
Figure 1.1
where
notation
i
=
1, 2,
x..
denotes the value of the function at the point
(x..)
,m,
(i, j).
is also used to represent the matrix. For each
the sequence
(x
il' x i2,
.
'
x.
)
The
6
is called the
ith
row of the matrix; for each
.th
column. The matrix
j =
1 ,
2,
..
,
n,
the
sequence
is called the
J
only if the function is one -to -one;
i.e. ,
is univalent if and
(x..)
13
x
=
implies
xid
(i,j) =(k,.)
In the next chapter we shall define a
rectangular arrangement
as a simple mathematical structure consisting of a set
binary relations R, C
(for "row" and "column" ),
certain conditions specified by the axioms
of
X
and two
satisfying
rectangular arrange-
ments. We then show that every univalent matrix determines a unique
rectangular arrangement
of
its elements. In Chapter III we shall
study the properties of rectangular arrangements and establish the
general relationship between these structures and matrices; namely,
that each rectangular arrangement that is not a single row or column
corresponds to four distinct matrices (depending
on which
corner
element is chosen for x11).
Note:
The term "mathematical structure"
place of "simple mathematical structure"
of
this paper, for the sake of brevity.
will be used in
throughout the remainder
7
CHAPTER II
THE MATHEMATICAL STRUCTURE
(X, R, C)
To set up the mathematical structure of a rectangular arrange-
ment, we take the set
consisting
X
ment and two binary relations
is as follows: The pair
R
x
is next to
belongs to
C
if and only if
=
y,
The interpretation of
X.
on
C
(x,y)
or
x
R,
the objects in the arrange-
of
belongs to
R
if and only if
in the same row. Similarly,
y
x
=
or
y,
is next to
x
(x, y)
in the
y
same column. The mathematical structure for this arrangement is
then denoted by
(X, R, C).
Before writing down the axioms for this structure, it is convenient to introduce four new relations
is interpreted as the set of all pairs
are in the same row, and ER
R
(x, y)
replaced by "column"
Definition
exists a sequence
(zi, zi} 1)eR
for
Definition
at most one
The pair
2. 1.
(z1, z2,
i
2. 2.
=
such that
and
x
Roo
y
,
and
Coo
is similar,
EC
.
(x,y)
belongs to
such that
zn)
x=
R
zl,
if there
y
=
zn
and
-1.
1, 2,
The object
y such that
ER, EC. Here
is a unary relation consisting of all
end -of -row objects. The interpretation of
with "row"
Coo,
y
x
x
belongs to
and
(x, y)cR.
ER
if there exists
8
Definition
exists a sequence
(z.,
1 z.
1)eC
for
Definition
The pair
2. 3.
(zl, z2,
,
zn)
1, 2,
,
n-
i
=
2. 4.
exists at most one
y
such that
x
=
C
z1,
y
if
there
=
zn
and
1.
The object
x
such that
y
The mathematical structure
belongs to
(x, y)
belongs to
#
and
x
EC
if
there
(x, y)eC.
is called a rectangular
(X, R, C)
arrangement if and only if the following axioms are satisfied for all
belonging to
x, y, u and v
X:
The interpretation of Axioms 2.
1
and
2. 2
is obvious: they will
be used without further explicit mention throughout the next chapter.
Axiom 2.
1.
The binary relations R,
C
on
are both
X
reflexive.
Axiom
2. 2.
The binary relations
R, C
on
X
are both
symmteric.
Axiom 2.3a ensures that for every object belonging to
X,
there are at most two objects next to it in the same row. Axiom
2. 3b
ensures the same property for columns.
Axiom 2. 3a.
z
-
x
and
There are at most two distinct
(z, x)eR.
z's
such that
9
z
4
x
and
z's
There are at most two distinct
Axiom 2. 3b.
such that
(z, x)EC .
Axiom
2. 4a
ensures that for every row of the arrangement there
is at least one end -of -row object. Similarly, by Axiom 2. 4b, for
every column there is at least one end -of- column object.
Axiom
2.
4a.
There exists a
zEER
such that
(x, z)ER00
.
Axiom
2. 4b.
There exists a
zEE
such that
(x, z)EC
.
We can
interpret Axiom
2. 5
co
as the assertion that each row
intersects each column in exactly one point (see Figure
2. 1).
Y
X
Figure
Axiom
and
(y, z)EC
2. 5.
Z
.
2.
1
There exists exactly one
z
such that
arrangement as in Figure
2. 2
where
(x, z)ER
.
oc
If we have an
are next to each other in the same row,
x
and
u
x and y
are next to
10
each other in the same column,
and
y
and
v
and
u
are in the same column,
in the same row and
y
is next to
are in the same row,
v
then
u
is next to
in the same column.
v
v
This
is implied by Axiom 2. 6.
x
y
u....
v
Figure.
Axiom 2. 6.
then
(u, v)ER
In
If
and
(x, u)cC,
(x, y)ER,
order to exclude the example
X
shall be concerned with finite sets
Axiom 2. 7.
We pause to
We
(u, v) »R
00
and
(y, v)EC
m
(y, v)cC.
jects, we require that the set
2. 7.
2. 2
The set
X
of an
arrangement without ob-
be non -empty.
of
In this paper we
objects.
is finite and non -empty.
consider the independence of Axioms
2.
1
through
shall illustrate their independence by giving an example
of
,
11
an arrangement that satisfies all but one of Axioms
2.
1
through 2.7.
The proof that the example satisfies all but the specified axiom is left
to the reader.
The mathematical structure with
and
C = {(a,
a), (b, b)
For the sake
of
X
=
{a, b
satisfies all but Axiom
}
(a, b)
that
and
We
r
belong to
R,
and
(a, b)cR
a
r
Figure
{(a, b), (b, a)
2. 1.
(a, a)
(a, b)
and
representation
and
(b, a)
R,
a
b
R
r
Figure
2. 3a
a
c
2. 3b
and
to denote
c
belong to C.
belong to
(b, b)
of the
to denote that
2. 3a
as in Figure
r
does not belong to
(b, a)
Figure 2.4 to denote that
understood that
as in Figure
shall use
(b, a)
=
brevity in presenting the remaining examples,
we introduce a figure notation to simplify the
arrangement.
R
},
and
as in
It is
C.
b
2. 3b
b
Figure 2.4
The arrangement in Figure 2.
5
satisfies all but Axiom
2. 2.
}
12
r
r
r
a
Figure
2.
5
It is interesting to note that Axiom 2.
is replaced by Axiom
2. 6
Axiom
then
If
If
2. 6 *.
and
(v, u)ER
that
(y, x)eR.
that
(u,x)EC.
as follows:
(x, y)ER,
x
=
u
(x, u)EC,
If
can be omitted if Axiom
(x, u)EC,
and
y
taking
and the arrangement as in Figure
y
=
u and
2. 6b
y
\
Figure
A
Axiom
x
c
/r
"circular arrangement" as in Figure
2. 4a
,
it follows
u
\
y
2. 4b.
2. 3b.
2. 7a
w
/
u
c
2. 6b
satisfies all but
and the "circular arrangement" as in Figure 2. 7b
satisfies all but Axiom
2.
cZ\c
Figure
2, 6a
=
satisfies all but Axiom
r
u
y
satisfies all but Axiom
z
r
r
(y, v)EC
it follows from this axiom
y
=
An arrangement as in Figure 2. 6a
x
and
(u, v)ER
(v, y)EC.
taking
(x, y)cR,
2. 6*
2
3a
13
r/yr
x
\
r
c/y\c
/
\//
\//
Figure
z
`
satisfy all but Axiom
r
2.
7b
2. 8,
will
r
r
f
e
c
C
c
C
r
d
Figure
2. 6.
c
arrangements, as in Figure
b
C
Axiom
c
2. 5.
a
2. 9
z
Figure
2. 7a
Two separate rectangular
Figure
x
h
2. 8
represents an arrangement that satisfies all but
(Take
x
=
a23, u = a32, y
r
r
a CI11 a l2 a
cI
C
c
r
r
a22, y
r
C
=
a21
a33
r
CI
a22
r
a23
2. 9
as represented in Figure
2. 7.
r
c
r
a21
r
a 12
I
in Axiom 2. 6)
3--f--a32
An "infinite arrangement"
a 11
a33
CI
Figure
satisfies all but Axiom
=
CI
r
a22
Figure
2. 10
2. 10
14
The proof that the following mathematical structures are rec-
tangular arrangements is left to the reader.
Example
ure
a, b,
Consider the row of objects
2. 1.
as in Fig-
c
2. 11.
r
a
r
b
Figure
2.
c
1
1.
The rectangular arrangement is given by the set
X
=
{a, b, c
together with the binary relations
(b, b),
(c , c ),
(b, a),
(a, b),
consisting of the pairs
2.
1
it follows that
the pairs
(a, c)
R
oc
and
2. 4
and
(c, a).
2. 3
of
,
all pairs
b),
(a, a),
and the relation
C
From Definition
of the
set
R
and also
By Definition 2. 2,
=
(a, c
}.
it follows that
that
EC
(c
and (c, c).
(a, a), (b, b)
ER
Similarly, by Definition
consisting of the pairs
R
(b, c)
consists
},
=
{a, b,
c
} .
Coo=
00
C
and by Definition
15
Example
Consider a univalent
2. 2.
matrix
mX n
(x..)
as
1J
given by Definition
1. 8
and illustrated in Figure
A
1. 1.
rectangular
arrangement is given by the set
-<i <m,
X= {xi. :1
together with the binary relations
R
and C,
xij), (xij, xij), (xin, xin):
{(xij, xi(J+ 1)), (xia+
=
R
1<j <n}
1
< i_< m,
1
<j
<n },
and
C = {(xi j, x( i +l )j), (x( i+1
)j,
x iJ'
iJ
xij),
i
m
x
:
j
1<i<m,
1<j<n}.
-
This mathematical structure is called the rectangular arrangement
associated with the univalent matrix (x..).
(X, R, C)
From Definition
R
0o
2.
=x
13
it follows that
1
i/ ) :
.
1
-< i -< m,
1
< j <
n,
< j <
n,
1
-<
< n
.
By Definition 2. 2,
ER=
From Definition
{x
x
,
it
2. 3
.
in
1
:
-< i <- m }.
it follows that
Coo= {(xij,
J
):
J
1
< i <
m,
1
1
<
Q
<
m
} .
16
By Definition 2.4,
E
=
{x
,
j
The main
x.:
1
< j < }.
J
result
of the next
chapter is that every rectangular
arrangement is associated in this way with some univalent matrix.
This result shows that our definition captures the essence of rec-
tangular arrangements alone, by excluding arrangements of all other
kinds.
17
CHAPTER III
SOME PROPERTIES OF RECTANGULAR
ARRANGEMENTS (X, R, C)
In this chapter we shall derive
trary rectangular arrangement
We
several properties
of an
arbi-
(X, R, C).
shall first show that each rectangular arrangement has a
dual, obtained by interchanging
R
and
which is again a rec-
C,
tangular arrangement.
Definition 3.
Let
1.
Then the structure
(X, R, C)
obtained by interchanging
(X, C, R)
is called the dual of
(X, R, C).
Theorem
The dual,
3. 1.
ment
(X, R, C)
when
R
and
changed with
Proof:
C
EC.
be a rectangular arrangement.
of a
(X, C, R),
Coo
ER
It follows
are interchanged,
is interchanged with
from Definitions
Roo
2.
1
4
is interchanged with
Moreover, it is easy to check that Axioms
if these interchanges are made in the axioms.
2. 2
-2.
Roo
that when
and
Coo,
2.
is inter-
EC.
1
-2.
6
2. 3b
and
R
ER
with
remain true
Thus, Axiom
are essentially unchanged, Axioms 2.3a and
C,
Moreover,
are interchanged to form the dual,
and
and
rectangular arrange-
is itself a rectangular arrangement.
C
R
2.
1
and
are interchanged,
18
as are Axioms 2.4a and
2.
"dummy variables"
and
x
4b; Axiom 2.
(X, R, C) is a
is unchanged provided the
are interchanged, and Axiom
y
unchanged if the "dummy variables"
It follows that if
5
y
and
u
2. 6 is
are interchanged.
rectangular arrangement, so also is
(X, C, R).
An important fact, closely related to Theorem 3. 1, is that each
proposition that can be proved in the theory
remains true if R,
and
Rco
and
ER
of
rectangular structures
are interchanged with C,
Coo
respectively. The new proposition obtained in this way will
EC,
be called the dual of the original proposition. To see that the dual of a
proposition
says the same thing about the dual structure
dual of
P
as
says about
P
P itself is true, notice that the
is true if and only if
P
(X, C, R)
(X, R, C).
Several lemmas and theorems of this chapter are arranged in
dual pairs, marked "a"
and "b"
such as Theorem 3.3a and b.
,
case only the "a" statement need be proved, since the "b"
In each
statement is its dual.
Theorem 3.
2.
The relations
R
and
C
are equivalence
relations.
Proof:
To prove that
R
co
is an equivalence relation, we
must show that it is reflexive, symmetric and transitive (see Definitions
1.3
to
1. 5).
19
For reflexivity, let a be any object in
be the sequence of length two such that
z
(a, a)ER; hence by Definition 2. 1, (a, a)ER
For symmetry,
Definition
znn
sequence
Then
i
wl
there is
(z., z.
..
b,
=
,
wn
n -1.
a sequence
i
=
w.
=
,n- 1.
zn -. 1,
and by Axiom 2. 2,
Hence
(b, a)ER
and
,
(w.,w.1+
1
there is
(v., vi+
1)ER
a sequence
for all
i
(v1, v2,
=
,
1, 2,
,
ul = b, um= c, (u., u.
such that
n-
1,
1)ER
.)
sider the sequence
w2,
(w
,
for
w.
ui-n+
Then
Hence
w
=
a,
(a, c)ER
j = 1,
wm +n -1
and
,
00
We have shown
assertion is that
C
00
=
R
that
and
c
00
R
1
By Definition
v1
a, vn= b,
=
,un)
2,
,
i=
1,2,,n -1
for
i =
n,n +1,.
(w.,w.
m- 1. Con-
such that
{
i
1)ER for
for
i
a,
,n.
i = 1, 2,
and a sequence (u1, u2,
wm +n -1)
v.
z 1=
Consider the
(b, c)ERco.
such that
vn)
By
.
is reflexive.
R
For transitivity, suppose (a, b)ERco and
2.
R
such that
zn)
,
l' z2)
is reflexive.
00
pair belonging to
1, 2,
such that
R
(z
By Axiom 2. 1,
a.
=
and
,
z2,
(z
for all
,wn)
a
=
be any
(a, b)
1)ER
(w1,w2,
1, 2,
=
1
and
b
=
2.
let
z2
=
and let
X,
)ER
,m +n -1
for i =1,2,
.
,m +n -2.
is transitive,
00
is an equivalence relation.
is an equivalence relation.
The dual
20
It is
II)
clear from the interpretation
of
Roo
and
C
00
(see Chapter
that the equivalence classes [1] of these two equivalence relations
are just the rows and columns
rectangular arrangement. This
of the
is the motivation for
Definition 3.
The equivalence classes of
2.
and
R
C
00
are
called rows and columns, respectively.
If an object is not at the end of a row, then
there must be two
objects next to it in the same row. This, and the dual statement for
columns, is the assertion
Theorem
of
Theorem 3.3a -b.
If
x
does not belong to
exist exactly two objects
z
such that
b.
3. 3a.
exactly two objects
Proof:
z
:x
and
that xCER.
If
z
such that
x
z
and
(z, x)eR,
by Definition 2.
2
we
(x, z)cR.
then there exist
(x,z)eC.
there are not at least two distinct
z's
such that
reach the contradiction
Thus, there are at least two such
2.3a there are at most two such
z's
EC,
does not belong to
x
If
# x and
z
then there
ER,
z's
and by Axiom
z's. Hence there are exactly two
and the theorem follows.
Out next theorem
asserts that
two objects which are in the same
row, and also in the same column, are necessarily equal.
21
Theorem
x
=
3. 4.
If
belongs to both
(x, y)
and
R
then
C
y.
By Theorem 3.2
Proof:
statement
(x, z)cR
,
co
(y, z)EC
(x, x)cR
and
00
is true for
co
(y, y)EC
z =
.
00
Hence the
and for
x
z
y,
=
but by Axiom
2. 5
there is exactly one such z. Thus
Lemma
3.1a
asserts that the columns containing two different
objects of a given row are disjoint.
Figure
(See
x
3. 1.
=
z
=
y.
)
z
w
Y
Figure 3.
Lemma 3. la.
(x, y)cRoo,
and
w
and both
(z, x)
be objects such that
x, y, w, z
and
(w, y)
x
Co Then
belong to
y,
z
are distinct.
b.
and both
Let
1
Let
(z, x) and
x, y, w, z
(w, y)
be
objects such that
belong to
Roo.
Then
x L y, (x, y)EC
z
and w
,
are
distinct.
Proof:
to
Coo.
Suppose that
z = w.
It follows from Theorem 3.
Then
2
that
(w, x)
(x, y)EC
and
.
00
(w, y)
belong
By Theorem 3. 4
22
x
=
contradicting the hypothesis that x
y,
y1
y.
Thus
and
z
w
must be distinct.
Theorem 3.
asserts that an object next to and in the same
5a
column as an end -of -row object is also an end -of -row object (see
Figure 3.
2).
Corollary 3. la asserts the same property for any object
in the same column as an end -of -row object.
X
x1
Y
Y
1
Figure 3.2
Theorem 3. 5a.
If
xEER
and
(x, y)EC,
then
yEER.
b.
If
xCEC
and
(x, y)ER,
then
ycEC.
x
y,
then obviously yEER.
Proof:
y
If
=
does not belong to
z's such that
objects
z1
and
z2.
x
y,
suppose
Then by Theorem 3. 3a there are two
ER.
distinct
If
y and
z
(See
(z, y)ER.
Figure 3.3.
u1
x
u2
zl
y
z2
Figure 3.3
)
We
shall call these
23
By Axiom 2.
5
there exists a ul
from Axiom
It follows
(u
If
u1
Theorem 3.
=
2
then
x,
(z
l,y)
such that
(x,
zl)
Since
such that
u2 # x,
z1 =z2,
xEER
and
(x, y)EC00
b.
If
xEEC
and
(x,
1
In
(u2, z2)EC.
that
u1
2
=
u2.
(zl, z2)ER00,
contradicting the fact that
If
(z.,z.
y,
Similarly,
x.
and
z 1=
are distinct. Hence yEER.
1)EC
for
i
=
(z
z2,
,
then
y)ER then
directly from Theorem
It follows
on the length of a sequence
and
3.4 that
(zl, z2)ECoo. Since by Theorem 3.
Corollary 3. la.
Proof:
2
and by
C,
u1
(x, u2)ER,
it follows from Definition 2.
xEER,
(ul,zl)EC.
and
belong to
Thus
y.
z1
it follows from Theorem 3.4 that
and z2
(x, y)
It follows from Theorem
EC00.
Thus by Theorem 3. 2,
z
(ul, x)ER
that
and
contradicting the assumption that
there exists a u2
2. 6
(zl, ul)EC00 and
3. 5a
zn) such that
yEER.
yEEC.
by induction
x=
y= zn
z
1,2,,n -1.
order to establish that each rectangular arrangement is asso-
ciated with some matrix, we introduce a special type of sequence.
Definition 3.3.
w.
e
=
w.
if and only if
sequence
A
i
=
j
for
(w1, w2,
i,
j = 1,
2,
,
wn)
such that
,n is called
a uni-
J
valent sequence.
The next theorem shows that we may always take the sequence
24
(z
1,
C
z2,
,
in Definition 2.
zn)
(x, y)ER
If
a unique univalent sequence
=
and
u
(ui, ui+
Proof:
one sequence
for
(u., u.1+11)ER
for
(x, y)ER
.
(u1, u2,
i
'
,
least element, which
such sequence
m
,
sequences is a nonempty set
a
for
and x
co
,
such that
un)
x
u1,
,
such that
un)
1, 2,
=
then there exists a
y,
f
,
x
=
ul,
n -1.
by Definition 2.
such that
x
=
there is at least
1
u1, y
=
m-1. The set of lengths
of
=
1,2,,n -1.
=
i
um)
1, 2,
=
,
(u1, u2,
1)EC
Since
i
(x,y)EC
If
unique univalent sequence
=
2. 3
then there exists
x # y,
and
oo
(u1, u2,
(u.,ui +1)ER for
and
un
b.
y
or in Definition
Roo,
to be univalent.
,
Theorem 3. 6a.
y
for
1
um
and
of
all such
positive integers. Therefore it has
shall denote by n. We shall show that every
we
(u1, u2,
,
un)
minimal length is a univalent se-
of
quence.
Suppose that
u.
=
u.
i
the sequence
and
vi=
such that
u.e
(v1, v2,
i+ j
for
x= V1, y
for some
,
.Q
=
vn -j
quc71.ce
)
where
j,
i, j
=
,n.
1, 2,
and (vj, vj+
(ù1,u2,
with this property; hence,
u.
i
for
vv = ui2
,n-j+i is
i+ 1, i+ 2,
vn- J +i
This contradicts the fact that
=
i <
Then
J
1)ER
,
for
i
=
1, 2,
,i,
shorter sequence
=
.
un) is the
u. for
J
a
.Q
j.
1,2,
, n- j +i
shortest se-
-J..
25
The proof of uniqueness depends on the following:
Proposition
such that n
There is no univalent sequence (u1, u2,
3. 1.
(u., u.
> 3,
for
FR
1)
i
1, 2,
=
...
n- 1 and (ul, un)
,
un)
. . . ,
E
R.
Proof: Suppose such a sequence (ul, u2, ... , un) exists. By
Axiom 2.4a there is a
such that (u1,
z EER
...
term of the sequence (u1, u2,
no
,
By definition 2. 2,
z) E.R09.
un) belongs to ER; hence,
is
z
not a term of this sequence. As proved above, there is a univalent
sequence (vi, v2,
for i
1, 2,
=
...
1, 2, ...
(ul, u2,
i
=
...
,
...
,
vr) such that vi
r -1. Since
un), there is a k,
,
,
n and
yk+
=
and
ui -1, ui+
un
=
u1
if i
+l
tion is proved.
=
n.
1
ul, yr
u1, and yr
=
=
1, 2,
...
,
n.. Then there are
where we take uo
=
un if i
=
and
... , wm)
for i
(w., w. 1)E.R
1, 2,
=
exists a greatest integer k,
-
If k
valent, n
=
=
n,
un
and
... , m -1.
=
wn
n
=
1
<
k
=
wm, and since (w1, w2,
> n.
=
+
1
<
.
<
=
wm
. . .
n and of
=
Then there
u, for all
wm) is uni-
m and the two sequences are equal. If k
the least integer such that k
wl, y
Since the sequence
n, such that w.
<
let
3. 6a,
be a univalent sequence such that x
(ul, u2, ... , un) is the shortest such sequence, m
i < k.
1,
Since this contradicts Axiom 2.4a, the proposi-
Returning to the proof of uniqueness in Theorem
(wl, w2,
ui for some
=
such that w f vk and (vk,w)ER; namely,
w
yk+ 1,
is not a term of
z
=
and (vi, vi +1)ß.R
z
=
r, such that vk
< k <
u.j for all j
11
1
three distinct values of
w
v1
=
< n,
let
w. for some
3
.Q
be
26
j
=
k
+ 1,
k
+ 2,
(uk, uk+
...
,
,
...
Consider the sequence
m.
,
'wj-1,...'wk+2'wk+1).
u
By the definition of k, we have
sequence is of length
> 3.
ukl
uk +1
hence, the
wk+l # uk;
Furthermore, by the definition of f, the
sequence is univalent. By Proposition 3.
1
no such sequence
exists.
This contradiction rules out the case k <n, completing the proof.
shall next show that a row with at least two objects has
We
exactly two distinct end -of -row objects.
Theorem
3. 7a.
If there exist distinct x, y such that (x, Witco,
then there exist exactly two distinct z'svER such that (x,
z)r_ R00.
b. If there exist distinct x, y such that (x, y)e C00,
then there exist exactly two distinct z's SE
such that (x,
z) S C00.
Proof: Since x / y and (x, y)F. R00, it follows from Theorem
3. 6a
that there exists at least one univalent sequence (u1, u2,
such that x
=
u. for some j
=
1, 2,
J
i
=
1, 2,
...
,
m-1. Since the set
X
...
,
m and (u., u.
3.
is finite (Axiom
)
2. 7),
...
,
um)
R for
the set of
lengths of all such sequences is a finite nonempty set of integers.
Therefore it has
a
From Definition
3. 3, u1 # un.
greatest element, which
We
we shall denote by n.
shall show that u
and un
belong to ER.
Suppose u1 does not belong to ER.
By Theorem 3. 3a
there
27
exist exactly two z's such that z#ul and (u1, z)ER. Let these z's be
denoted by zl and z2. Since u2 $ u1 and (u1, u2)ER, it follows from
that u 2= zi for i= or i= 2. We may assume without loss
of generality that z2 = u2. We will show that (z1, u u2, , , , un) is a uniAxiom
2. 3a
1
valent sequence, contradicting the fact that (u1, u2,
...
,
un) is the
longest such sequence.
Suppose that z
=
the univalent sequence (zl, u, u2,
for i=
3.
1
1, 2;
and since
no such sequence
t ui for
uk, and z
z1
=
...
i
uk- 1),
,
=
1, 2,
. , . ,
k -1.
Now k >3 since zl
uk we have (z1, uk -1)ER.
t ui
By Proposition
exists. Thus no such k exists and
is a univalent sequence.
Consider
(z1 , u1,
... , un)
This contradiction establishes that ujIER.
Using a similar argument, it follows that unf ER.
To prove that u1 and un
are the only two such objects, we
need the following proposition:
Proposition
3. 2.
such that (ui, ui )`R for i
+1
=
,
,
If z is any object such that (z,
to ER.
some
... un)
1, 2, ... n -1
Let (ul, u2,
j
=
1,2,
be a univalent sequence
and both ul and un belong
ul)ER, then
z
=
u, for
... ,n.
Proof: Follows by an argument similar to that used in the first
part of the proof of Proposition
3, 1,
Returning to the proof of the theorem, suppose ER contains
a
third object
sition
3. 2 z
z
such that
=u. for some i
not belong to ER.
u1
t
=
2, 3,
z #
...
un and (u1, z)ERco.
,
n-1. By Definition
By Propo2. 2
ui does
This contradicts the assumption that zEER, and
completes the proof.
If an object is both an end -of -row and an end -of- column, we
28
shall call it a "corner object"
We show in
.
Theorem 3. 8 that every
rectangular arrangement that is not the special case
of a
single row
or column has four distinct corner objects. Before we show the
existence
of four such
objects, we shall show that there is at least
one corner object for each pair of objects, where one of the pair is
an end -of -row and the other an end -of- column.
Lemma 3. 2.
zeER
(EC such that
Proof:
and
zEEC.
zeER
If
(y,
Theorem 3.
Roo
oo
Proof:
5
ycEC,
zeER
there is a
8.
Coo,
0o
We see
If
r,EC,
E
R
z
zeER,
such that
(x, z)EC
co
and by Corollary 3. lb
and the lemma follows.
there exists
then
there exists a
(z, y)cR
and
Corollary 3. la
By
Therefore,
belong to
(z, x)eCoo
By Axiom 2.
z)eR.
and
cm
a
E
C
pair
(x, y)
which does not
contains exactly four objects.
that the hypothesis does not allow the arrange-
ment to be a single row or column. As a guide in the proof we may
visualize the arrangement as in Figure 3.4.
29
yl
z
1
.
z3
.
Y
x1
z
x
.
.
.
2
w
x2
Y2
z4
Figure 3.
By Axiom 2.
(w, y)EC
.
co
Since
(x, y)
from Theorem 3.
2
and
w
x1
and
distinct
(x2, x)
y1 and y2
(y2, y) belong to
the following
C00.
a
such that
w
does not belong to
that
there are distinct
(x1, x)
there exists
5
4
# x,
x2
belong to
and
w
belonging to
or
C
R00,
# y. By Theorem
belonging to
R00.
(x, w)ER
ER
such that
It follows from Lemma 3.
2
it follows
3. 7a
such that
By Theorem 3. 7b
EC
and
there are
(y1, y) and
that there exist
z's:
(i)
z iEER
SEC
such that
(z
(ii)
z2eER
nEC
such that
(z2, y2)ER
yl)ER00
and
(z
x1)EC00
.
and
(z2, x1)EC00
.
30
(iii)
z3EER'mEC
such that
(z3, y1)ERo
and
(z3,x2)e Coo
(iv)
z4EER,-EC
such that
(z4, y2)ERco
and
(z4, x2)ECco
We see
that
z
Since
(y1,y2)ER0.
for
f z 2;
zk # zß
for
that
Now suppose
from
k,1
1,
k
(z1, u1)ECco
(u, u2)ECco.
and
and
ER EC
5.
y1
=
contra-
y2,
Similarly, it follows that
contains a fifth point
there exists a
5
and a
(u, ul)ERco,
(See Figure 3.
y2.
3.4
2
1,2,3,4.
By Axiom 2.
z2, z3, z4.
z
=
then by Theorem 3.
= z
z
by Theorem
(y1,y2)ECco,
dicting the distinctness of y1
if
u2
distinct
such that
u1
such that
u,
u2)
(z
ERoo
and
)
u2
z
1
u
u1
Figure 3.
By
Corollary
3. lb,
ul
If
=
u1
z1
u1
=
la,
and
or
ul
=
2,
and u2
u1
belong to
u2
then
z1,
Theorem 3.
3.
z2,
(z
5
belong to
EC.
Thus by Theorem
and by Theorem 3.7a
u2)
(u2, u)ER.
and
(z
and by Corollary
ER
u)
u2
=
belong to
z1
3. 7b,
or
Roo
u2= z3.
and by
It follows from Theorem 3. 4 that
u2= u.
31
This contradicts the supposition that
u
so that
# z1,
u1
=
u1
z2.
=
zi for
u
Similarly,
u2
contradicting the assumption that
z4,
=
i
=
1,2,3,4. Thus
and by Axiom 2.
z3
zl,
is distinct from
u
5
Thus the theorem follows.
z2, z3, z4.
Theorem 3.9 implies that
a
rectangular arrangement with one
object has one corner object, and a rectangular arrangement having
one row or one column and at least two objects has exactly two
corner
objects.
Theorem 3.
fied; i, e.
If the conditions
9.
every pair
,
for Theorem 3.
y)ERUC,
(x,
are not satis-.
8
then ERS EC
contains one
or two objects.
Proof:
Definition
set
If the
2. 2
X
contains exactly one object
and by Definition
zeER
ERA EC.
belongs to
If the
set
contains at least two objects, let
X
two distinct objects. Then
(x, y)ER
(x, y)
(x, y)EC00
but not both. Suppose that
Cog
does not belong to
Let
w
does not belong to
Roo,
does not belong to
Rog
Co
By
Theorem 3.
2
uC 00
R rThC
00
.
00
(x, y)eR00
Thus
it follows from Theorem 3.
(x, y)EC00,
(w, x)
and
does not
(x, y)
X.
2
or
(x, y)ER
and
be any object belonging to
Thus both
be any
x, y
It follows from Theorem
.
00
3.4 that
belong to
then by
Thus one object
zeEC.
2. 4
z,
that
(w, y)
If
(x, w)
(w,y)
belong to
contradicting the supposition that
32
Thus (x, w)ER . Hence, there is
o
one row. It follows from Theorem 3. 7a that there exist exactly two
(x, y)
does not belong to
u's belonging to
ER.
If
follows from Theorem 3.
Thus
uEER EC.
Coo.
there exists a
that
4
u
=
y.
such that
y
Thus by Definition 2. 4
ucEC.
there are
ER,
Since only two objects belong to
it
(y, u)cC,
exactly two corner objects.
Similarly,
if
and
(x,y)EC00
does not belong to
(x, y)
Roo,
there is one column with two corner objects.
Figure 3.
6
illustrates Theorem
z
w
x
y
Figure 3.
Theorem 3.
C
,
and
1Oa.
(z, w)ER
,
b.
R
,
and
(z, w)EC
Proof:
,
If
then
ul,
By Axiom 2.
5
there exists
(y, w)
belong to
1
and
z = un
a
(x, z)
and
belong to
(y, w)
(z,w)EC.
By Definition 2.
=
and
(x, z)
(z,w)ER.
(x, y)EC,
If
x
such t h a t
6
(x, y)ER,
then
3. 1Oa.
v2
there exists
(ui' ui+ l)EC
such that
a sequence (u1, u2,
for
i
=
1, 2,
(v2, u2)ERco
and
un)
,
n -1.
33
and by Axiom 2.
(v2, y)ECco,
By _ontinuing in this way,
(v2, v3,
i ä
that
,n-l.
2, 3,
u
v
=
If a row
(vn, un)eR
and
(v2, y)eC.
sequence
(vi, vi+
1)cC
for
It follows from Definition 2.3 and Theorem 3.
and for
n
Thus
u.
we obtain a
Hence the statement
(y, v )eC .
n
00
true for
such
such that
vn)
,
and
(v2, u7)eR
6
u
=
v
=
n
u
=
w.
and
00
By Axiom 2.
Hence
w.
(z, u)cR
5
(y, u)eC
00
2
is
there is exactly one
(z,w)eR.
has at least two distinct objects, and one object in the
row is a corner object, then there exists another object in the same
row that is a corner object.
Lemma 3. 3a.
and
xeERn EC.
xeERn EC.
and
(x, z)eC00,
(x, z)eRoo
and
Therefore,
only one
Let
x
y, (x, y)eR
such that
z
be objects such that
Then there exists a unique
z
00
# x,
x # y, (x, y)EC
such that
z
z
# x,
zeERn EC.
zeER.
zeERn EC.
z
x, y
By Theorem 3. 7a
Proof:
be objects such that
Then there exists a unique
b.
and
x, y
zeERn EC.
and
(x, z)eR00,
Let
This is the assertion of Lemma 3.3a.
there exists a
z
such that
It follows from Corollary 3. lb that
Furthermore, by Theorem
satisfying the lemma.
3. 7a,
z
zeEC.
there is
x,
34
Lemma 3. 4a asserts that any row with at least two objects can
be arranged in a univalent sequence in exactly two ways.
Lemma 3. 4a.
there exist distinct x,y
If
such that
then there exist exactly two distinct univalent sequences
such that
x
for some
u.
=
j
=
,n,
1, 2,
ul
both
(x, y)ER
(u1,
u2,
and
,
,
un)
un
J
belong to
ER,
and
then
if (z, x)ER
u.
(u., 1+1
for
)ER
for some
z = u.
i
1, 2,
=
,
moreover,
n -1;
J.
J
b.
x,y such that
there exist distinct
If
then there exist exactly two distinct univalent sequences
such that
x
for some
u.
=
,n,
j = 1, 2,
both
(x, y)EC
(u1,
u2,
,
.,un)
u1
and
n -1;
moreover,
un
J
belong to
and
E
(z,x)EC00 then
if
m
z2.
z
=
It follows by
sequence
i
=
1, 2,
i
=
1, 2,
.
=
z1,
,
there are exactly two distinct
un
=
z2
and
Proposition 3,2 that x
,wn)
,n. Then
w1
n -1,
1, 2,
z's
=
(u.,
1 u.
=
such that
1)ER
for
for some
u.
w.
=
un -i+
i
j.
1
wn
=
z1,
the two univalent sequences are different.
and
,u1
(u1, u2,
2,
= 1,
,
-1.
Consider the
for
(w.,w. 1)ER
and the sequence is univalent. Since
z2,
zl
be denoted by
there exists a univalent sequence
(wl,w2,
,
=
J
Let the
(x, z)ER00.
By Theorem 3. 6a
u
i
for some
u.
By Theorem 3. 7a
such that
such that
for
1)EC
3
Proof:
z'sEER
(u., ui+
for
w
ul
35
Now suppose that
(v1, v2,
such that
vm)
,
there exists a third univalent sequence
x
=
for some
v.
1, 2,
j =
»
m,
,
J
both
.
i = 1, 2,
z 2=
and
vm
and
v1
,
belong to
vm,
z 1=
or
v.
and
n.
for
u.
=
1
1
i
=
vm
2. 2
and
that v2
n
=
=
=
u2
If
v
that
z
v 1= z
u2,
( u1
vl
=
=
it
u
1
and from Axiom 2. 3a
,k where k is the smaller
1, 2,
Since the sequences
are univalent,
z2
for
(vi, v. 1)cR
m -1. It follows from Theorem 3. 7a
follows from Definition
that
and
ER
,un)
and
( v1
of
v2,
m
,
v)
This implies that the two sequences are the
m.
same, contradicting the assumption that they are different. Thus
z2
=
m
=
and
v1
n
and
z1
w.
i
=
=
Using a similar argument, it follows that
vm.
for
v.
i
=
1, 2, '
.
,n,
contradicting the assump-
tion that they are different. Hence there exist exactly two such
sequences.
If a
(xi.),
rectangular arrangement is associated with a univalent matrix
we shall show that
x11 has to be one of the
corner objects.
J
Once this corner object has been picked, this matrix is uniquely
determined. This is the assertion
Lemma 3.
5.
If a
of the following
rectangular arrangement
ated witha univalent matrix (xi.),
then
lemma.
(X, R, C)
x11cERfm EC.
If a
is associ-
rectan-
J
gular arrangement is associated with two univalent matrices (x..),(w..)
13
1J
with
x11
- w11
then
(x..)
J
=
(w..).
J
36
It is easy to verify by Example 2.
Proof:
Suppose the matrices
2
are
(x..) and (w..)
xl
that
1EER r
Ec.
dimension
of
J
m1X n1
and
n1
It follows from Definition 2.
n2.
induction that
loss
(x
In
of
,
x
respectively. Let n be the lesser
m2X n2,
and
w
=
lj
x
generality that
1(n -1)
the fact that
Similarly,
and
)
n2
(x
In
,
>
w
n1.
1(n +1)
2.
1
3a,
< i <
m,
by induction on
we have
xij
belong to
)
2. 2).
=
wij
contradicting
n1
and
m1
then
1)'
R,
Thus
by
assume without
We
there exists a
m be the lesser of
that the matrices (x..)
and (w..)
13
13
i,
If
and Axiom 2. 3a
,n.
j = 1, 2,
(see Example
x1nEER
letting
for
lj
2
of
=
n2
=
n.
it follows
m2,
are both mX n. For each fixed
using Definition 2.
j
for all j,
1
< j <
n.
2
and Axiom
Thus
(xij) = (wij)
and the lemma follows.
In Chapter II (Example 2. 2) we saw that a
rectangular arrange-
ment is associated with every univalent matrix. The question we now
consider is: For every rectangular arrangement
a
mX n
ment
univalent matrix
(X, R, C)
(x..)
(X, R, C)
is there
such that the rectangular arrange-
is associated with this matrix
(x..)
13
?
The answer
to this question is yes; in fact for every rectangular arrangement
there is one, two or four distinct matrices, depending on the number
of
corner objects. This will be proved in the next theorem.
Theorem 3.
11.
Let
(X, R, C)
be a rectangular arrangement
37
with
n
corner objects. Then
is associated with exactly
(X, R, C)
univalent matrices.
n
From Theorem 3.
Proof:
and 3.
8
9
to consider the following four cases. Case
gular arrangement with one object,
it is obviously sufficient
1
will consider the rectan-
Case 2a will consider the rec-
tangular arrangement having one row such that the set contains at
least two objects, Case
will consider the rectangular arrange-
2b
ment having one column such that the set contains at least two objects
and Case
will consider the rectangular arrangement having at least
3
two rows and two columns.
Case
the set
X
1.
Let
be a rectangular arrangement such that
(X, R, C)
contains exactly one object. This rectangular arrange-
ment is associated with exactly one matrix; namely, the one -by -one
matrix (x11),
Case 2a.
where
Let
is the single member of
x11
(X, R, C)
only one row, and such that
X
X.
be a rectangular arrangement having
contains at least two objects. By
Lemma 3. 4a this row is the range of either of two distinct univalent
sequences. Let one of these be denoted
1X n
matrix
(x)
with
xlj=ui
for
(u1, u2,
j = 1, 2,
,un). Define the
,
n.
Using
J
Example
1
X
n
2. 2,
matrix
the rectangular arrangement associated with this
.)
J
is the set
38
X= {xl.J :
and the two binary relations
R
=
and
R
1
< j < n
},
C,
{(xlj,xl(j +1))' (xl(j+ 1)' x1j), (xlj,xlj), (xln'xln):
< j <
1
n},
and
C = {(x
,
lj
x
lj ):
1
< j <
Thus we see that
with this
1X n
n
}
.
is the rectangular arrangement associated
(X,R,C)
matrix
(x1 ).
J
Similarly, define the
1X n
matrix (x1).) with
x1
=
j
for
j = 1, 2,
' ' '
,n. It follows that
arrangement associated with this
un-
j
+
1
(X,R,C) is also the rectangular
1X n
matrix
(x1 ).
j
The two matrices are different since their one -one elements
are different. By Lemma 3.
5
there exist at most two such matrices.
Thus each row is associated with two distinct univalent matrices.
Case 2b.
Let
(X,R,C)
only one column and such that
be a rectangular arrangement having
X
contains at least two objects.
Using an argument similar to that used in Case 2a, it follows that this
rectangular arrangement is associated with two distinct matrices
coinciding with the two univalent sequences of Lemma 3. 4b.
39
Let
Case 3.
be a rectangular arrangement having
(X, R, C)
at least two rows and two columns. By Theorems 3.
8
exist four distinct elements belonging to ERr.EC.
Let one of these
w, y, (ul, u2,
be denoted by z. Let
-
and
un)
,
and 3.
(v1, v2,
,
-
there
9
vm)
be defined as follows:
1)
we
ERr.EC
2)
and
and
(y, z)eC
(ul, u2,
,
3. 6a such that
is the univalent sequence given by Theorem
un)
ul,
z =
(vl, v2,
w
,
=
un
vl,
For fixed i, j,
1
z
object given by Axiom
y
- m,
-
< j <
1
n,
1
< j <
such that
(xi., xij )ER
Theorem 3. l0a that
for
1
< i <
(x(i+ 1)j' xij)
m,
1
1<.
<
n,
J
and
1
belong to
n.
C
By
for
<
00
x..
-
,m -1.
be the unique
< i <
(xi(j+ 1)'
m,
v.)cR
(x..,
13
1
00
Thus it follows from
Theorem 3. 10b,
1
and
for
(x..,
)e:C
13 xkj
00
k < m.
and
(xij, xi(j+ 1))
< j <
let
(x.., u.)eC
1J
n,
,n -L
1J
J
<i <m,
j = 1, 2,
J
l)eC for i= 1, 2,vm and (v.,v.
1
=
< i <
2. 5
By Theorem 3. 2,
for
(u., u.+ l)eR
and
is the univalent sequence given by Theorem
vm)
=
3. 6b such that
1
such that y # z,
.
J
4)
z,
w
(w, z)eRco
is the object given by Lemma 3. 3b
y
ycER SEC
3)
is the object given by Lemma 3.3a such that
w
1
xij)
belong to
(xij, x(i+ 1)j)
< j <
n.
R
and
By Axiom
.
40
2. 1,
(x..,
x..)cR
13
13
Corollary 3. la,
il and
x.
by Corollary 3. lb,
for
(x..,
x..)cC
1J
13
and
x1
Now suppose that
in
x..
1J
.
.
=
xm
x
.
.
.
1
u.
for
By
-< i -< m
1
for
EC
us
.
< j < n.
1
rs . (See Figure 3. 7.
1
< j <
and
n.
)
n
s
x.
lj
v
.
r
v
ER
belong to
j
J
v.
- - m,
< i <
belong to
x.
and
j
1
x
rs
m
Figure 3.7
By Axiom 2. 2,
(v.,
v )ER
1
r
sequence
00
.
(x..,
x )ER. It follows from Theorem 3.
1J
rs
By Theorem 3. 4,
(v1, v2,
it follows that
u.
,
=
v)
us,
v.
1
=
v
is univalent.
=
x
rs
if and only if
i
=
(r, s)
=
(i, j).
r
that
since the
Using a similar argument,
which is true if and only if
J
x..1J
r ; hence,
2
j =
s. Thus
41
(xil' xrs)ER for (r, s) 4 (i, 1), (r, s) (i, 2), then by
Definition 2. 2 xil does not belong to ER, contradicting the fact
If
that
xil EE R
.
then by Definition
the fact that
(xi
If
xrs)ER
2. 2
x.
x. EER'
If
in
(r, s) # (i,n -1),
(r, s) i (i,n),
for
(xi. , xrs)ER
for
1
contradicting
ER,
does not belong to
m,
< i <
1
< j <
n,
J
(r, s) # (i, j),
(r, s)
(i, j+ 1),
(r, s) # (i, j -1),
Axiom
is
2. 3a
contradicted. Thus it follows that
R
=
{(xij,
(xi(j+ 1), x..), (xij, xij), (x in xin ):
x. (J+ 1)),
1
m,
< i <
1
< j < n }.
Using a similar argument, it follows that
C
=
{(x..iJ ,
x
1)j
1
)
(
x(i+
):
x.),
(x
, x
x ), (x..,
iJ
iJ
mj mj
1)j' ij
associated with at least one
and
i
z4
1, 2, 3, 4.
=
< i <
-
m,'
< j < n }.
Thus we see that a rectangular arrangement
z3
1
mX n
univalent matrix.
be the four corner objects.
Take
Then the rectangular arrangement
ated with these four
mX n
is
(X, R, C)
xll
Let
=
zi
(X, R, C)
z1, z2,
for
is
associ-
univalent matrices. The four matrices
are distinct since their one -one elements are different. By Lemma
3.
5
there exist at most four such matrices. Thus the theorem follows.
42
With this theorem we have completed the association of rectan-
gular arrangements with univalent matrices.
43
CHAPTER IV
EXTENSIONS OF THE CONCEPT OF
RECTANGULAR ARRANGEMENTS
Two dimensional rectangular arrangements can be generalized
further by allowing the set
property of Axiom
2. 7.
X to be
infinite; i. e. , deleting the finite
In this case rectangular arrangements can
still be associated with univalent matrices except that the matrices
may be infinite. If there is only one corner object, then the set has
exactly one element or the set is infinite. If the set is infinite, then
either the rectangular arrangement is a row as in Figure
column, or an arrangement as in Figure 4.
all
a13
a12
Figure
all
a12
a21
a22
4.
Figure 4.
1
2
2.
4. 1,
or a
44
If
there are two corner objects and the set is finite, we have
a single
row or single column. If the set is infinite, then we have an arrange-
ment as in Figure 4. 3a or Figure 4. 3b.
all
a12
all
a12
aln
a21
a22
a21
a22
a2n
aml
amt
Figure
If
Figure
4. 3a
4. 3b
there are more than two corner objects, then there exist four cor-
ner objects and Axiom
Figures
2. 7
4. 4 and 4.
5
follows.
typify infinite arrangements that are not
included in this generalization.
al -1
a10
all
alt
a2-1
a20
a21
a22
am -1
amo
aml
amt
Figure
4. 4
45
a-1-1
a-10
a-11
a-12
a0-1
0-1
a00
a01
a02
al-1
a10
all
a12
Figure
We have
mension.
4.
5
considered only rectangular arrangements
It seems that with
of two di-
appropriate relations it would be pos-
sible to capture in mathematical terms rectangular arrangements
of
di-
mension n. As an example it seems that a three dimensional rectangu-
lar arrangement as in Figure
relations,
R
and
C
,/t112
I
//'1212
212
a211
I
//
ra121
a221
a312
a311
could be captured with three binary
as given and a third binary relation
"next to in depth".
alll
4. 6
r122
I
1222
I
/13
ja232
a131
a231
a33 2
a3 22
a3
a321
Figure
a331
4. 6
2
D
for
46
We have seen that it is
lations the essence
of a
possible to capture with two binary re-
rectangular arrangement. Can
arrangement be captured with one binary relation?
that a more abstract kind
of
a
rectangular
It would
seem
rectangular arrangement could be pre-
sented with a single binary relation N, for "next to"
.
In this
abstraction the distinction between rows and columns would be lost.
It seems that this
rectangular arrangement would be associated with
one, four or eight matrices, depending on the number of corner ob-
jects. Further work is being done.