CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES

The Pure and Applied Mathematics.
2014.
May,
21(2):
121-128

- Received : March 01, 2014
- Accepted : May 01, 2014
- Published : May 31, 2014

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The Boolean rank of a nonzero
m
×
n
Boolean matrix
A
is the least integer
k
such that there are an
m
×
k
Boolean matrix
B
and a
k
×
n
Boolean matrix
C
with
A = BC
. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank
k
with 1 ≤
k
≤ min{
m,n
}.
binary Boolean algebra
consists of the set
= {0, 1} equipped with two binary operations, addition and multiplication. The operations are defined as usual except that 1 + 1 = 1.
Let
denote the set of all
m
×
n
Boolean matrices with entries in
. The usual definitions for adding and multiplying matrices apply to Boolean matrices as well. Throughout this paper, we shall adopt the convention that 3 ≤
m
≤
n
unless otherwise specified.
The (
Boolean
)
rank, b(A)
, of nonzero
A
∈
is the least integer
k
such that there are Boolean matrices
B
∈
and
C
∈
with
A
=
BC
. It follows that 1 ≤
b(A)
≤
m
for all nonzero
A
∈
. The Boolean rank of the zero Boolean matrix is 0.
A mapping
T
:
→
is called a
linear operator
if
T
(
αA
+
βB
) =
αT(A)
+
βT(B)
for all
A,B
∈
and for all
α,β
∈
.
A linear operator
T
on
is called a (
P,Q
)-
operator
if there are permutation matrices
P
and
Q
of orders
m
and
n
, respectively, such that
T(X)
=
PXQ
for all
X
, or
m
=
n
and
T(X)
=
PX^{t}Q
for all
X
, where
X^{t}
is the transpose of
X
.
Let 1 ≤
k
≤
m
. For a linear operator
T
on
, we say that
Beasley and Pullman (
[1]
) have characterized linear operators on
that preserve Boolean rank as follows:
Theorem 1.1.
For a linear operator T on
,
the following are equivalent
:
The characterization of linear operators on vector space of matrices which leave functions, sets or relations invariant began over a century ago when in 1897 Fröbenius
[7]
characterized the linear operators that leave the determinant function invariant. Since then, several researchers have investigated the preservers of nearly every func- tion, set and relation on matrices over fields. See
[6
,
7]
for an excellent survey of Linear Preserver Problems through 2001. For Boolean matrix and Boolean rank are important research topics on matrix theory. See
[4
,
5]
for detailed contents and applications on Boolean matrix theory.
Recently Beasley and Song (
[3]
) have obtained a new characterization of Theorem 1.1: For a linear operator
T
on
,
T
preserves Boolean rank if and only if
T
preserves Boolean ranks 1 and
k
, where 1 ＜
k
≤
m
. They also have obtained characterizations of the linear transformations that preserve term rank between different matrix spaces over semirings containing the binary Boolean algebra in
[2]
.
In this paper, we extend Theorem 1.1 to any two consecutive Boolean rank preservers. Furthermore we obtain other characterizations.
O
is an arbitrary zero matrix and
J_{m,n}
is the
m
×
n
matrix all of whose entries are 1. A matrix in
is called a
cell
if it has exactly one 1 entry. We denote the cell whose one 1 entry is in the
(i, j)^{th}
position by
E_{i,j}
. Further we let
ε_{m,n}
be the set of all cells in
. That is,
ε_{m,n}
= {
E_{i,j}
| 1 ≤
i
≤
m
, 1 ≤
j
≤
n
}.
If
A
and
B
are Boolean matrices in
, we say that
A dominates B
(written
B
⊑
A
or
A
⊒
B
) if
a_{i,j}
= 0 implies
b_{i,j}
= 0 for all
i
and
j
. This provides a reflexive and transitive relation on
. For Boolean matrices
A
and
B
in
with
B
⊑
A
, we define
A
＼
B
to be the Boolean matrix
C
such that
c_{i,j}
= 1 if and only if
a_{i,j}
= 1 and
b_{i,j}
= 0 for all
i
and
j
.
Lemma 2.1
(
[1]
).
If T is a linear operator on
,
then T is invertible if and only if T permutes ε_{m,n}
.
A Boolean matrix
L
∈
is called a
line matrix
if
L
=
or
L
=
for some
i
∈ {1, …,
m
} or for some
j
∈ {1, …,
n
}:
R_{i}
=
is the
i
th
row matrix
and
C_{j}
=
is the
j
th
column matrix
.
For a linear operator
T
on
, we say that
T preserves line matrices
if
T(L)
is a line matrix for every line matrix
L
.
Lemma 2.2.
Let T be an invertible linear operator on
.
Then T preserves line matrices if and only if T is a (P,Q)-operator
.
Proof
. By Lemma 2.1,
T
permutes
ε_{m,n}
and hence
T(J_{m,n})
=
J_{m,n}
. Let
T
preserve all line matrices. Now we will claim that either
If
m
≠
n
, (1) is satisfied since
T
is invertible and preserves all line matrices.
Thus we assume that
m = n
. Suppose that the claim is not true. Then there are distinct row matrices
R_{i}
and
R_{j}
(or column matrices
C_{i}
and
C_{j}
) such that
T(R_{i})
is a row matrix and
T(R_{j})
is a column matrix. But then
T(J_{m,n})
=
T(R_{1})
+⋯
T(R_{i})
+⋯+
T(R_{j})
+⋯+
T(R_{n})
cannot dominate
J_{m,n}
. This contradicts
T(J_{m,n})
=
J_{m,n}
. Hence the claim is true.
Case (1): We note that
T(R_{i})
=
R_{α(i)}
for all
i
and
T(C_{j})
=
C_{β(j)}
for all
j
, where
α
and
β
are permutations of {1, …,
m
} and {1, …,
n
}, respectively. Then for any cell
E_{i,j}
, we have
T(E_{i,j})
=
E_{α(i),β(j)}
. Let
P
and
Q
be the permutation matrices corresponding to
α
and
β
, respectively. Then for any Boolean matrix
we have
Hence
T
is a (
P,Q
)-operator.
Case (2): We note that
m = n
,
T(R_{i}) = C_{α(i)}
for all
i
and
T(C_{j}) = R_{β(j)}
for all
j
, where
α
and
β
are permutations of {1, ⋯,
n
}. By a parallel argument similar to Case (1), we obtain that
T(X)
is of the form
T(X) = PX^{t}Q
, and thus
T
is a (
P,Q
)-operator. The converse is obvious. □
For nonzero
A
∈
, it is well known (
[1]
) that
b(A)
is the least integer
k
such that
A
is the sum of
k
Boolean matrices of Boolean rank 1. This establishes the following:
Lemma 2.3.
For Boolean matrices A and B in
,
we have
b (A + B ) ≤ b(A) + b(B) .
Theorem 2.4.
Let T be an invertible linear operator on
and
1 ≤
k
≤
m
.
Then T preserves Boolean rank k if and only if T is a (P;Q)-operator
.
Proof
. By Lemma 2.1,
T
permutes
ε_{m,n}
. Assume that
T
preserves Boolean rank
k
. Now, we will show that
T
preserves line matrices, and then
T
is a (
P
,
Q
)-operator by Lemma 2.2. For the case of
k
= 1, it is clear that
T
preserves line matrices since the Boolean rank of every line matrix is 1. Thus we assume that
k
≥ 2. Suppose that
T
does not preserve a line matrix. Then there are two distinct cells
E_{i,j}
and
E_{s,t}
that are not dominated by the same line matrix such that
T(E_{i,j})
and
T(E_{s,t})
are dominated by the same line matrix. Without loss of generality, we assume that
T
(
E
_{1,1}
+
E
_{2,2}
) =
E
_{1,1}
+
E
_{1,2}
. So, we have a contradiction for the case of
k
= 2. Hence we assume that
k
≥ 3. Then for the Boolean matrix
A
=
E
_{3,3}
+ ⋯ +
E_{k,k}
, we have
b
(
E
_{1,1}
+
E
_{2,2}
+
A
) =
k
. But by Lemma 2.3,
b (T (E _{1,1} + E _{2,2} + A )) ≤ b (T (E _{1,1} + E _{2,2})) + b(T(A)) ≤ 1 + (k − 2) = k − 1,
a contradiction to the fact that
T
preserves Boolean rank
k
. Hence
T
preserves line matrices. The converse is obvious. □
T
on
is
singular
if
T(X)
=
O
for some nonzero
X
∈
; otherwise
T
is
nonsingular
. In fact, if
T
is a singular linear operator on
, then we can easily check that
T(E) = O
for some cell
E
. Further, if
T
is a (
P,Q
)-operator on
, then
T
is nonsingular.
Example 3.1.
For 1 ≤
k
≤
m
, let
A
=
E
_{1,1}
+
E
_{2,2}
+ ⋯ +
E_{k,k}
2
. Define an operator
T
on
by
T(O) = O
and
T(X) = A
for all nonzero
X
2
. Clearly,
T
is linear, nonsingular and preserves Boolean rank
k
, while
T
does not preserve Boolean rank.
The number of nonzero entries of a Boolean matrix
A
∈
is denoted by #(
A
).
Lemma 3.2.
Let
1 ≤
k
＜
m and
1 ≤ l ≤
m
.
Assume that T is a linear operator on
.
If
then T is nonsingular.
Proof
. If
T
is singular, then
T(E) = O
for some cell
E
. Hence we have a contradiction for the case of
k
=
l
= 1. Thus we assume that
k, l
≥ 2. Now, choose Boolean matrices
A
and
B
with
E
⊑
A
and
E
⊑
B
such that
b(A)
= #(
A
) =
k
+ 1 and
b(B)
= #(
B
) =
l
. It follows that
b
(
A
＼
E
) =
k
and
b
(
B
＼
E
) =
l
− 1. But then
T(A)
=
T
(
A
＼
E
) +
T(E)
=
T
(
A
＼
E
) contradicts the condition (i) and
T(B)
=
T
(
B
＼
E
) +
T(E)
=
T
(
B
＼
E
) contradicts the condition (ii). Hence
T
is nonsingular. □
Lemma 3.3.
Let T be a linear operator on
.
If
then T maps cells to cells.
Proof
. If
T
preserves Boolean rank
k
and
k
+ 1 with 1 ≤
k
≤
m
− 1, or
T
strongly preserves Boolean rank
k
with 1 ≤
k
≤
m
, then
T
is nonsingular by Lemma 3.2. Suppose that
T
does not map cells to cells, in particular suppose that
T(E)
dominates two cells for some cell
E
. By permuting we may assume that
T(E)
⊒
E
+
F
for some cell
F
≠
E
.
If
E
and
F
are in the same row, we may assume by permuting that
E
=
E
_{1,k+1}
and
F
=
E
_{1,k}
. If
E
and
F
are in the same column, we may assume by permuting that
E
=
E
_{k+1,1}
and
F
=
E
_{k,1}
. If
E
and
F
are in different rows and different columns, we may assume by permuting that
E
=
E
_{1,k+1}
and
F
=
E
_{2,k−1}
. For 1 ≤
r
≤
m
, let
W_{r}
= [
] where
= 0 if and only if
i
+
j
≤
r
. Then
b(W_{r})
=
r
. Since
E
⊑
W
_{k+1}
and
F
⋢
W
_{k+1}
, we have that
b
(
W
_{k+1}
+
E
) =
k
+1 and
b
(
W
_{k+1}
+
F
) =
k
.
Let
L = T^{d}
where
d
is chosen so that
L
is idempotent (
L
^{2}
=
L
). Then,
L
preserves Boolean ranks
k
and
k
+ 1 for case (i), or
L
strongly preserves Boolean rank
k
for case (ii) and
L(E)
⊒
E
+
F
.
Now, since
L(E)
=
F
+
X
for some Boolean matrix
X
,
L(E) + F = (X + F ) + F = X + F = L(E)
and since
L
is idempotent,
L(E) = L ^{2}(E ) = L(L(E)) = L (L(E) + F )
= L ^{2}(E ) + L(F) = L(E) + L(F) = L (E + F ).
That is,
L
(
E
+
F
) =
L(E)
. Thus if
Y
is any Boolean matrix which dominates
E
, we have that
L
(
Y
+
F
) =
L(Y)
since
L
(
Y
+
F
) =
L
(
Y
+
E
+
F
) =
L(Y)
+
L
(
E
+
F
) =
L(Y)
+
L(E)
=
L
(
Y
+
E
) =
L(Y)
. Thus,
L (W _{k+1}) = L (W _{k+1} + F ).
However,
b
(
W
_{k+1}
) =
k
+ 1,
b
(
W
_{k+1}
+
F
) =
k
and
L
preserves both Boolean rank
k
and
k
+1 for case (i) or
L
strongly preserves Boolean rank
k
for case (ii). Thus, we have
k + 1 = b (L (W _{k+1})) = b (L (W _{k+1} + F )) = k ,
which is a contradiction for the both cases (i) and (ii). Therefore
T
maps cells to cells. □
Theorem 3.4.
Let T be a linear operator on
.
Then T preserves Boolean rank if and only if
Proof
. Let
T
preserve Boolean ranks
k
and
k
+1 or
T
strongly preserves Boolean rank
k
. Then
T
maps cells to cells by Lemma 3.3. Now, suppose that
T
is not invertible. Then
T(E)
=
T(F)
for some distinct cells
E
and
F
by Lemma 2.1. If
b
(
E
+
F
) = 2, choose a Boolean matrix
A
∈
with
b(A)
= #(
A
) =
k
−1 such that
b
(
E
+
A
) =
k
and
b
(
E
+
F
+
A
) =
k
+ 1. But then
k
+ 1 =
b
(
T
(
E
+
F
+
A
)) =
b
(
T
(
E
+
A
)) =
k
, a contradiction for both cases (i) and (ii). For the case of
b
(
E
+
F
) = 1, we may assume, without loss of generality, that
E
=
E
_{1,1}
and
F
=
E
_{1,2}
. Let
B
=
E
_{2,1}
+
E
_{2,2}
+
E
_{3,3}
+ ⋯ +
E
_{k+1,k+1}
. Then
b
(
E
+
F
+
B
) =
k
and
b
(
E
+
B
) =
k
+ 1. But then
k
=
b
(
T
(
E
+
F
+
B
)) =
b
(
T
(
E
+
B
)) =
k
+ 1, a contradiction for both cases (i) and (ii). Thus
T
is invertible. By Theorem 2.4,
T
is a (
P, Q
)-operator and hence
T
preserves Boolean rank by Theorem 1.1. The converse is obvious. □
Recently Beasley and Song (
[3]
) showed that for a linear operator
T
on
,
T
preserves Boolean rank if and only if
T
preserves Boolean ranks 1 and
k
, where 2 ≤
k
≤
m
.
Now we summarize our results by:
Theorem 3.5.
Let T be a linear operator on
.
Then the following are equivalent
:
As a concluding remark, we suggest to prove the following conjecture:
Conjecture 3.6.
Let T be a linear operator on
.
Then T preserves Boolean rank if and only if T preserves any two Boolean ranks h and k with
1 ≤
h
＜
k
≤
m
≤
n
.

1. INTRODUCTION

The
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- (1)T preserves Boolean rank kifb(T(X)) = kwheneverb(X) = kfor allX;
- (2)T strongly preserves Boolean rank kif,b(T(X)) = kif and only ifb(X) = kfor allX;
- (3)T preserves Boolean rankif it preserves Boolean rankkfor allk∈ {1, 2, …,m}.

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- (i)T preserves Boolean rank;
- (ii)T preserves Boolean ranks1and2;
- (iii)T is a (P,Q)-operator.

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2. PRELIMINARIES

The matrix
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- (1)Tmaps {R1, …,Rm} onto {R1, …,Rm} and maps {C1, …,Cn} onto {C1, …,Cn}, or
- (2)Tmaps {R1, …,Rn} onto {C1, …,Cn} and maps {C1, …,Cn} onto {R1, …,Rn}.

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3. CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS

An operator
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- (i)T preserves Boolean rank k and k+ 1,or
- (ii)T strongly preserves Boolean rank l,

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- (i)T preserves Boolean rank k and k+ 1with1 ≤k≤m− 1,or
- (ii)T strongly preserves Boolean rank k with1 ≤k≤m,

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- (i)T preserves Boolean rank k andk+ 1with1 ≤k≤m− 1,or
- (ii)T strongly preserves Boolean rank k with1 ≤k≤m.

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- (i)T preserves Boolean rank;
- (ii)T preserves Boolean ranks k andk+ 1,where1 ≤k≤m− 1;
- (iii)T preserves Boolean ranks1and k, where2 ≤k≤m;
- (iv)T strongly preserves Boolean rank k, where1 ≤k≤m;
- (v)T is a (P,Q)-operator.

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Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(No. 2012R1A1A2042193).

Beasley LeRoy B.
,
Pullman Norman J.
1984
Boolean rank preserving operators and Boolean rank-1 spaces
Linear Algebra and Its Applications
59
55 -
77
** DOI : 10.1016/0024-3795(84)90158-7**

Beasley L.B.
,
Song S.Z.
2013
Linear transformations that preserve term rank between different matrix spaces
J. Korean Math. Sci. Soc
50
(1)
127 -
136
** DOI : 10.4134/JKMS.2013.50.1.127**

Beasley L.B.
,
Song S.Z.
2013
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63
435 -
440
** DOI : 10.1007/s10587-013-0027-z**

deCaen D.
,
Gregory D.
,
Pullman N.
1981
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Proceedings of the Third Caribbean Conference on Combinatorics and Computing
Barbados
169 -
173

Kim K.H.
1982
Boolean Matrix Theory and Applications
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New York

Li C.-K.
,
Pierce S.J.
2001
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Amer. Math. Monthly
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(7)
591 -
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** DOI : 10.2307/2695268**

Pierce S.J.
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33
1 -
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** DOI : 10.1080/03081089208818176**

Citing 'CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES
'

@article{ SHGHCX_2014_v21n2_121}
,title={CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES}
,volume={2}
, url={http://dx.doi.org/10.7468/jksmeb.2014.21.2.121}, DOI={10.7468/jksmeb.2014.21.2.121}
, number= {2}
, journal={The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={BEASLEY, LEROY B.
and
KANG, KYUNG-TAE
and
SONG, SEOK-ZUN}
, year={2014}
, month={May}