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Linear operators preserving commuting pairs of matrices over semirings.

[section]1. Introduction and preliminaries

A semiring means an algebra (S, +, *, 0, 1), where + and * are binary, 0 and 1 are nullary, satisfying the following conditions:

(1) (S, +, 0) is a commutative monoid;

(2) (S, -, 1) is a monoid;

(3) a * (b + c) = a * b + a * c and (a + b) * c = a * c + b * c, [??]a, b, c [member of] S;

(4) a * 0 = 0 * a = 0, [??]a [member of] S;

(5) 0 [not equal to] 1.

Let S be a semiring and [M.sub.n](S) the set of all n x n matrices over S. I is the identity matrix, and O is the zero matrix. Define + and * on [M.sub.n](S) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to verify that <[M.sub.n](S), +, *, O, I> is a semiring with the above operations. Let [phi] be an operator on [M.sub.n](S). Then we say that [phi] preserves commuting pairs of matrices if [phi](A)[phi](B) = [phi](B)[phi](A) whenever AB = BA.

It is well known that commutativity of matrices is very important in the theory of matrices. Many authors have studied linear operators that preserve commuting pairs of matrices over fields and semirings. Song and Beasley [9] gave characterizations of linear operators that preserve commuting pairs of matrices over the nonnegative reals. Watkins [14] considered the same problem on matrices over the field of characteristic 0. Moreover, there are papers on linear operators that preserve commuting pairs of matrices over some other special fields (see [1, 2, 8]). In [10], Song and Kang characterized such linear operators over general Boolean algebras and chain semirings.

Linear preserver problem (LPP for short) is one of the most active research areas in matrix theory. It concerns the classification of linear operators that preserve certain functions, relations, subsets, etc., invariant. Although the linear operators concerned are mostly linear operators on matrix spaces over fields or rings, the same problem has been extended to matrices over various semirings (see [3, 5, 6, 7, 9, 10, 13] and references therein).

In the last decades, there are a series of literature on linear operators that preserve invariants of matrices over a given semiring. Idempotent preservers were investigated by Song, Kang and Beasley [11], Dolzan and Oblak [3] and Orel [6]. Nilpotent preservers were discussed by Song, Kang and Jun [13] and Li and Tan [5]. Regularity preservers were studied by Song, Kang, Jun, Beasley and Sze in [4] and [12]. Pshenitsyna [7] considered invertibility preservers. Song and Kang [10] studied commuting pairs of matrices preservers.

In this paper we study linear operators [phi] on the matrices over the direct product of copies (need not be finite) of a binary Boolean algebra such that [phi] preserves commuting pairs of matrices.

For convenience, we use [Z.sup.+] to denote the set of all positive integers.

Hereafter, S will always denote an arbitrary semiring unless otherwise specified.

Let A be a matrix in [M.sub.n](S). We denote entry of A in the ith row and jth column by [a.sub.ij] and transpose of A by [A.sup.t].

For any A [member of] [M.sub.n](S) and any [lambda] [member of] S, define

[lambda]A = [[lambda][a.sub.ij]]nxn, A[lambda] = [[a.sub.ij][lambda]]nxn.

A mapping [phi]: [M.sub.n](S) [right arrow] [M.sub.n](S) is called a linear operator (see [9]) if

[phi](aA + Bb) = a[phi](A) + [phi](B)b

for all a, b [member of] S and all A, B [member of] [M.sub.n](S).

Given k [member of] [Z.sup.+]. Let [B.sub.k] be the power set of a k-element set [S.sub.k] and [[sigma].sub.1], [[sigma].sub.2], *** , [[sigma].sub.k] the singleton subsets of [S.sub.k]. We denote [phi] by 0 and [S.sub.k] by 1. Define + and * on [M.sub.n](S) by

A + B = A [union] b, A * B = A [intersection] B.

Then <[B.sub.k], +, *, 0, 1> becomes a semiring and is called a general Boolean algebra (see [13]). In particular, if k = 1 then [B.sub.1] is called the binary Boolean algebra (see [13]).

A matrix A [member of] [M.sub.n](S) is said to be invertible (see [10]) if there exists B [member of] [M.sub.n](S) such that AB = BA = I. A matrix P [member of] [M.sub.n](S) is called a permutation matrix (see [3]) if it has exactly one entry 1 in each row and each column and 0's elsewhere. If P [member of] [M.sub.n](S) is a permutation matrix, then [PP.sup.t] = [P.sup.t]P = I. Note that the only invertible matrices in [M.sub.n]([B.sub.1]) are permutation matrices (see [10]).

[section]2. Main results

Let [([S.sub.[lambda]]).sub.[lambda][member of][conjunction]] be a family of semirings and S = [[PI].sub.[lambda][member of][conjunction]] [S.sub.[lambda]]. For any [lambda] [member of] [conjunction] and any a [member of] S, we use [a.sub.[lambda]] to denote [a.sub.([lambda])]. Define

[(a + b).sub.[lambda]] = [a.sub.[lambda]] + [b.sub.[lambda]], [(ab).sub.[lambda]] = [a.sub.[lambda]][b.sub.[lambda]] (a, b [member of] S, [lambda] [member of] [conjunction]).

It is routine to check that <S, +, *, 0, 1> is a semiring under the above operations. For any A = [[a.sub.ij]] [member of] [M.sub.n](S) and any [lambda] [member of] [conjunction], [A.sub.[lambda]] : = [[([a.sub.ij]).sub.[lambda]]] [member of] [M.sub.n]([S.sub.[lambda]]). It is obvious that

[(A + B).sub.[lambda]] = [A.sub.[lambda]] + [B.sub.[lambda]], [(AB).sub.[lambda]] = [A.sub.[lambda]][B.sub.[lambda]] and [(aA).sub.[lambda]] = [a.sub.[lambda]][A.sub.[lambda]]

for all A, B [member of] [M.sub.n](S) and all a [member of] S.

Hereafter, S = [[PI].sub.[lambda][member of][conjunction]], where [S.sub.[lambda]] is a semiring for any [lambda] [member of] [conjunction]. In the following, we can easily obtain

Lemma 2.1. Let A and B be matrices in [M.sub.n](S). Then the following statements hold:

(i) A = B if and only if [A.sub.[lambda]] = [B.sub.[lambda]] for all [lambda] [member of] [conjunction];

(ii) AB = BA if and only if [A.sub.[lambda]][B.sub.[lambda]] = [B.sub.[lambda]][A.sub.[lambda]] for all [lambda] [member of] [conjunction];

(iii) A is invertible if and only if [A.sub.[lambda]] is invertible for all [lambda] [member of] [conjunction]. The following result is due to Orel [6].

Lemma 2.2. If [phi] : [M.sub.n](S) [right arrow] [M.sub.n](S) is a linear operator, then for any [lambda] [member of] [conjunction], there exists a unique linear operator [[phi].sub.[lambda]] : [M.sub.n]([S.sub.[lambda]]) [right arrow] [M.sub.n]([S.sub.[lambda]]) such that ([phi][(A)).sub.[lambda]] = [phi][lambda]([A.sub.[lambda]]) for all A [member of] [M.sub.n](S).

Now, let [phi] a linear operator on [M.sub.n](S). Suppose that [[phi].sub.[lambda]] preserves commuting pairs of matrices for all [lambda] [member of] [conjunction]. For any A, B [member of] S, if AB = BA, then for any [lambda] [member of] [conjunction], [A.sub.[lambda]][B.sub.[lambda]] = [B.sub.[lambda]][A.sub.[lambda]]. This implies that

[[phi].sub.[lambda]]([A.sub.[lambda]])[[phi].sub.[lambda]]([B.sub.[lambda]]) = [[phi].sub.[lambda]]([B.sub.[lambda]])[[phi].sub.[lambda]]([A.sub.[lambda]]).

It follows that

[([phi](A)[phi](B)).sub.[lambda]] = [([phi](A)).sub.[lambda]][([phi](B)).sub.[lambda]] = [[phi].sub.[lambda]]([A.sub.[lambda]])[[phi].sub.[lambda]] ([B.sub.[lambda]]) = [[phi].sub.[lambda]]([B.sub.[lambda]])[[phi].sub.[lambda]]([A.sub.[lambda]]) = [([phi](B)).sub.[lambda]][([phi](A)).sub.[lambda]] = [([phi](B)[phi](A)).sub.[lambda]].

Further, [phi](A)[phi](B) = [phi](B)[phi](A). Thus [phi] preserves commuting pairs of matrices.

Conversely, assume that [phi] preserves commuting pairs of matrices. For any [lambda] [member of] [conjunction] and A, B [member of] [M.sub.n]([S.sub.[lambda]]), there exist X, Y [member of] [M.sub.n](S) such that [X.sub.[lambda]] = A, [Y.sub.[lambda]] = B and [X.sub.[micro]] = [Y.sub.[micro]] = O for any [micro] [not equal to] [lambda]. If AB = BA then XY = YX. Since [phi] preserves commuting pairs of matrices, we have [phi](X)[phi](Y) = [phi](Y)[phi](X). It follows that [[phi].sub.[lambda]](A)[[phi].sub.[lambda]](B) = [([phi](X)).sub.[lambda]][([phi](Y)).sub.[lambda]] = [([phi](X)[phi](Y)).sub.[lambda]] = [([phi](Y)[phi](X)).sub.[lambda]] = ([[phi](Y)).sub.[lambda]][([phi](X)).sub.[lambda]] = [[phi].sub.[lambda]](B)[[phi].sub.[lambda]](A). Hence [[phi].sub.[lambda]] preserves commuting pairs of matrices. In fact, we have proved

Lemma 2.3. Let [phi] : [M.sub.n](S) [right arrow] [M.sub.n](S) be a linear operator. Then [phi] preserves commuting pairs of matrices if and only if [[phi].sub.[lambda]] preserves commuting pairs of matrices for all [lambda] [member of] [conjuction].

Let [phi] be a linear operator on [M.sub.n](S). Suppose that [phi] is invertible. For any [lambda] [member of] [conjunction] and any A, B [member of] [M.sub.n]([S.sub.[lambda]]), there exist [bar.A], [bar.B] [member of] [M.sub.n](S) such that [([bar.A]).sub.[lambda]] = A, [([bar.B]).sub.[lambda]] = B, and [([bar.A]).sub.[micro]] = [([bar.B]).sub.[micro]] = O for any [micro] [not equal to] [lambda]. If [[phi].sub.[lambda]](A) = [[phi].sub.[lambda]](B) then

[([phi]([bar.A])).sub.[lambda]] = [[phi].sub.[lambda]](A) = [[phi].sub.[lambda]] (B) = [([phi]([bar.B])).sub.[lambda]].

Also,

([phi][([bar.A])).sub.[micro]] = [([phi]([bar.B])).sub.[micro]] = [[phi].sub.[micro]](O) = O

for any [micro] [not equal to] [lambda]. This shows that [phi]([bar.A]) = [phi]([bar.B]). Since [phi] is injective, we have [bar.A] = [bar.B]. Further,

A = [([bar.A]).sub.[lambda]] = [([bar.B]).sub.[lambda]] = B.

Thus [[phi].sub.[lambda]] is injective.

On the other hand, since [phi] is surjective, it follows that there exists X [member of] [M.sub.n](S) such that [phi](X) = [bar.B]. We can deduce that

B = [([bar.B]).sub.[lambda]] = [([phi](X)).sub.[lambda]] = [[phi].sub.[lambda]]([X.sub.[lambda]]).

That is to say, [[phi].sub.[lambda]] is surjective. Hence [[phi].sub.[lambda]] is invertible. We have therefore established half of Lemma 2.4. Let [phi] be a linear operator on [M.sub.n](S). Then p is invertible if and only if [[phi].sub.[lambda]] is invertible for all [lambda] [member of] [conjunction].

Proof. We only need to prove the remaining half. Assume that [[phi].sub.[lambda]] is invertible for any [lambda] [member of] [conjunction]. For any A, B [member of] [M.sub.n](S) and any [lambda] [member of] [conjunction], if [phi](A) = [phi](B) then

[[phi].sub.[lambda]] ([A.sub.[lambda]]) = [([phi](A)).sub.[lambda]] = [([phi](B)).sub.[lambda]] = [[phi].sub.[lambda]]([B.sub.[lambda]]).

Since [[phi].sub.[lambda]] is injective, we have [A.sub.[lambda]] = [B.sub.[lambda]]. By Lemma 2.1(i) it follows that A = B. Hence [phi] is injective. Since [[phi].sub.[lambda]] is surjective, there exists [X.sup.([lambda])] [member of] [M.sub.n]([S.sub.[lambda]]) such that [[phi].sub.[lambda]]([X.sup.([lambda])]) = [B.sub.[lambda]]. Let X [member of] [M.sub.n](S) satisfy [X.sub.[lambda]] = [X.sup.([lambda])] for any [lambda] [member of] [conjunction]. It is clear that [phi](X) = B, and so [phi] is surjective.

Thus [phi] is invertible as required.

The following lemma, due to Song and Kang [10], characterize invertible linear operators preserveing commuting pairs of matrices over a binary Boolean algebra.

Lemma 2.5. Let [phi] be a linear operator on [M.sub.n]([B.sub.1]). Then [phi] is an invertible linear operator that preserves commuting pairs of matrices if and only if there exists a permutation matrix P [member of] [M.sub.n]([B.sub.1]) such that either

(a) [phi](X) = [PXP.sup.t] for all X [member of] [M.sub.n]([B.sub.1]), or

(b) [phi](X) = [PX.sup.t][P.sup.t] for all X [member of] [M.sub.n]([B.sub.1]).

Next, we have

Theorem 2.1. Let S = [[PI].sub.[lambda][member of][conjunction]] [S.sub.[lambda]], where [S.sub.[lambda]] = [B.sub.1] for any [lambda] [member of] [conjunction]. Let [phi] : [M.sub.n](S) [right arrow] [M.sub.n](S) be a linear operator. Then [phi] is an invertible linear operator that preserves commuting pairs of matrices if and only if there exist invertible matrix U [member of] [M.sub.n](S) and [f.sub.1], [f.sub.2] [member of] S such that

[phi](X) = U ([f.sub.1]X + [[f.sub.2]X.sup.t])[U.sup.t]

for all X [member of] [M.sub.n](S), where [([f.sub.1]).sub.[lambda]] = [([f.sub.2]).sub.[lambda]] for any [lambda] [member of] [conjunction]. Proof. ([right arrow]) It follows from Lemma 2.3 and Lemma 2.4 that [[phi].sub.[lambda]] is an invertible linear operator which preserves commuting pairs of matrices for any [lambda] [member of] [conjunction]. By Lemma 2.5, there exists U [member of] [M.sub.n](S) such that either

[[phi].sub.[lambda]](X) = [U.sub.[lambda]][XU.sup.t.sub.[lambda]] (1)

for all X [member of] [M.sub.n]([S.sub.[lambda]]), or

[[phi].sub.[lambda]](X) = [U.sub.[lambda]][X.sup.t][U.sup.t.sub.[lambda]] (2)

for all X [member of] [M.sub.n]([S.sub.[lambda]]). Moreover, [U.sub.[lambda]] is a permutation matrix for any [lambda] [member of] [conjunction]. We have by Lemma 2.1 (iii) that U is invertible. Let [[conjunction].sub.i] : = {[lambda] [member of] [conjunction]| [[phi].sub.[lambda]] is the form of (i)}, i = 1, 2. It is evident that [[conjunction].sub.1] [intersection] [[conjunction].sub.2] = [??], [[conjunction].sub.1] [union] [[conjunction].sub.2] = [conjunction]. For i = 1, 2, let [f.sub.i] [member of] S satisfy [([f.sub.i]).sub.[lambda]] = 1 if [lambda] [member of] [[conjunction].sub.i] and 0 otherwise. We conclude that [phi](X) = U([f.sub.1]X + [f.sub.2][X.sup.t])[U.sup.t] for all X [member of] [M.sub.n](S) as required.

([left arrow]) For any [lambda] [member of] [conjunction] and any X [member of] [M.sub.n]([S.sub.[lambda]]), there exists Y [member of] [M.sub.n](S) such that X = [Y.sub.[lambda]]. We have

[[phi].sub.[lambda]](X) = [[phi].sub.[lambda]]([Y.sub.[lambda]]) = [([phi](Y)).sub.[lambda]] = [(U([f.sub.1]Y + [f.sub.2][Y.sup.t])[U.sup.t]).sub.[lambda]].

If [([f.sub.1]).sub.[lambda]] = 1, [([f.sub.2]).sub.[lambda]] = 0, then [[phi].sub.[lambda]](X) = [U.sub.[lambda]][XU.sup.t.sub.[lambda]] for all X [member of] [M.sub.n]([S.sub.[lambda]]). Otherwise, [[phi].sub.[lambda]](X) = [U.sub.[lambda]][X.sup.t.sub.[lambda]][U.sup.t.sub.[lambda]] for all X [member of] [M.sub.n]([S.sub.[lambda]]). We have by Lemma 2.1 (iii) that [U.sub.[lambda]] is invertible. It follows from Theorem 2.1 that [[phi].sub.[lambda]] is an invertible linear operator that preserves commuting pairs of matrices. Therefore, [phi] is an invertible linear operator that preserves commuting pairs of matrices by Lemma 2.3 and Lemma 2.4.

Song and Kang [10] characterize invertible linear operators which preserve commuting pairs of matrices over general Boolean algebra. Recall that a general Boolean algebra is isomorphic to a direct product of binary Boolean algebras. Up to isomorphism, we obtain characterization of invertible linear operators which preserve commuting pairs of matrices over general Boolean algebra.

Example 2.1. Let S = [B.sub.1] x [B.sub.1] x [B.sub.1]. Take

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in [M.sub.3](S) and [f.sub.1] = (0,1,0), [f.sub.2] = (1, 0,1) in S. Define an operator [phi] on [M.sub.3](S) by

[phi](X) = U ([f.sub.1]X + [f.sub.2][X.sup.t])[U.sup.t]

for all X [member of] [M.sub.3](S).

It is obvious that [U.sub.[lambda]]([lambda] = 1 , 2, 3) are all permutation matrices. By Theorem 2.1, [phi] is an invertible linear operator which preserves commuting pairs of matrices.

Example 2.2. Let S = [[PI].sub.k[member of]Z] + [S.sub.k], where [S.sub.k] = [B.sub.1] for any k [member of] [Z.sup.+]. Take a, b [member of] S, where [a.sub.2k-1] = 1, [a.sub.2k] = 0, [b.sub.2k-1] = 0 and b2k = 1 for any k [member of] [Z.sup.+]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

be a matrix in [M.sub.3](S). Define an operator [phi] on [M.sub.3](S) by

[phi](X) = U (aX + b[X.sup.t])[U.sup.t]

for all X [member of] [M.sub.3](S).

We have by Theorem 2.1 that [phi] is an invertible linear operator which preserves commuting pairs of matrices.

References

[1] L. B. Beasley, Linear transformations on matrices: the invariance of commuting pairs of matrices, 6(1978/79), No. 3, 179-183.

[2] M. D. Choi, A. A. Jafarian and H. Radjavi, Linear maps preserving commutativity, Linear Algebra Appl., 87(1976), 227-241.

[3] D. Dolzan and P. Oblak, Idempotent matrices over antirings, Linear Algebra Appl., 431(2009), No. 5-7, 823-832.

[4] K. T. Kang, S. Z. Song and Y. B. Jun, Linear operators that strongly preserve regularity of fuzzy matrices, Math. Commun., 15(2010), No. 1, 243-254.

[5] H. H. Li and Y. J. Tan, Linear operators that strongly preserves nilpotent matrices over antinegative semirings, Northeast. Math. J., 23(2007), No. 1, 71-86.

[6] M. Orel, Nonbijective idempotents preservers over semirings, J. Korean Math. Soc., 47(2010), No. 4, 805-818.

[7] O. A. Pshenitsyna, Maps preserving invertibility of matrices over semirings, Russ. Math. Surv., 64(2009), No. 1, 162-164.

[8] H. Radjavi, Commtativity-preserving operators on symmetric matrices, Linear Algebra Appl., 61(1984), 219-224.

[9] S. Z. Song and L. B. Beasley, Linear operators that preserve commuting pairs of nonnegative real matrices, Linear Multilinear Algebra, 51(2003), No. 3, 279-283.

[10] S. Z. Song and K. T. Kang, Linear maps that preserve commuting pairs of matrices over general Boolean algebra, J. Korean Math. Soc., 43(2006), No. 1, 77-86.

[11] S. Z. Song, K. T. Kang and L. B. Beasley, Idempotent matrix preservers over Boolean algebras, J. Korean Math. Soc., 44(2007), No. 1, 169-178.

[12] S. Z. Song, K. T. Kang, L. B. Beasley and N. S. Sze, Regular matrices and their strong preservers over semirings, Linear Algebra Appl., 429(2008), 209-223.

[13] S. Z. Song, K. T. Kang and Y. B. Jun, Linear preservers of Boolean nilpotent matrices, J. Korean Math. Soc., 43(2006), No. 3, 539-552.

[14] W. Watkins, Linear maps that preserve commuting pairs of matrices, Linear Algebra Appl., 14(1976), No. 1, 29-35.

(1) This work is supported by a grant of the science foundation of Northwest University# NC0925 and a grant of the science foundation of Shaanxi Province # 2011JQ1017.

Miaomiao Ren ([dagger]), Yizhi Chen ([double dagger]) and Guanwei Qin #

Department of Mathematics, Northwest University, Xi'an,

Shaanxi 710127, P. R. China

E-mail: mralgebra@sohu.com yizhichen1980@126.com qinguanwei11@163.com
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Date:Dec 1, 2011
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