Term rank preservers of bisymmetric matrices over semirings

In this article, we introduce another standard form of linear preservers. This new standard form provides characterizations of the linear transformations on the set of bisymmetric matrices with zero diagonal and zero antidiagonal over antinegative semirings without zero divisors which preserve some sort of term ranks and preserve the matrix that can be determined as the greatest one. The numbers of all possible linear transformations satisfying each condition are also obtained. Subjects: Science; Mathematics & Statistics; Advanced Mathematics; Algebra; Linear & Multilinear Algebra


Introduction
Linear preserver problems (LPPs) are one of the most active research topics in matrix theory during the past half-century, which have been studied when linear transformations on spaces of matrices leave certain conditions invariant. An excellent reference for LPPs is Pierce et al. (1992). There are many works on LPPs over various algebraic structures. The spaces of matrices over semirings also have been one of them.
Rank preserver problems play a pivotal role in investigating questions regarding other preservers. It can be found in Young & Choi (2008) that there are many nonequivalent definitions of rank ABOUT THE AUTHOR L. Sassanapitax is a PhD student in the Department of Mathematics and Computer Science, Chulalongkorn University. Sajee Pianskool and Apirat Siraworakun are lecturers at Chulalongkorn University and Thepsatri Rajabhat University, respectively. Our research interests are algebra and matrix theory.

PUBLIC INTEREST STATEMENT
A Boolean matrix is a matrix whose entries are 0 or 1 with addition and multiplication defined as if 0 and 1 are integers, except that 1 + 1 = 1. The term rank of a Boolean matrix A is the minimum number of lines containing all nonzero entries of A. Recently, the study of linear transformations preserving some certain functions over sets of many kinds of matrices has become established. In this article, we provide the patterns of linear transformations preserving term ranks of bisymmetric Boolean matrices via our new standard form. The number of such linear preservers is also counted. Moreover, the results on bisymmetric Boolean matrices are extended to bisymmetric matrices over semirings. We believe that the standard form that we have cooked up will be beneficial and will give a new point of view for research in related areas.
functions for matrices over semirings. Among many essential different definitions of rank functions of matrices over semirings, the combinatorial approach leads to the term ranks of such matrices.
Inspired by Beasley, Song and Kang's recently work, in Beasley, Song, & Kang (2012), on term rank preservers of symmetric matrices with zero diagonal over commutative antinegative semirings with no zero divisors, we investigate linear transformations on bisymmetric matrices whose all diagonal and antidiagonal entries are zeroes over such semirings that preserve some term ranks. We refer to Zhao, Hu, & Zhang (2008) and the references therein for more results and applications of bisymmetric matrices.
The significant survey about LPPs in Chapter 22 of Hogben (2007) indicates that linear preservers often have the standard forms. It turns out that our linear preservers do not possess any former standard forms; however, we invent a new standard form in order to obtain a natural and intrinsic characterization of term rank preserver on bisymmetric matrices whose all diagonal and antidiagonal entries are zeroes over commutative antinegative semirings with no zero divisors.
We organize this article as follows. In Section 2, some of the well-known terminologies and results on LPPs are reviewed and the notations in our work are introduced. In the third section, the results on term rank preservers of bisymmetric Boolean matrices with zero diagonal and zero antidiagonal are presented. Then, we extend the results to the case that all entries of such matrices are in commutative antinegative semirings with no zero divisors in the last section.

Definitions and preliminaries
We begin this section with the definition of a semiring. See Golan (1999) for more information about semirings and their properties.
Definition 2.1. A semiring S ; þ; Á ð Þis a set S with two binary operations, addition ( þ ) and multiplication ( Á ), such that: Þis a commutative monoid (the identity is denoted by 0); (ii). S ; Á ð Þ is a semigroup (the identity, if exists, is denoted by 1); (iii). multiplication is distributive over addition on both sides; We say that S ; þ; Á ð Þis a commutative semiring if ðS ; ÁÞ is a commutative semigroup. A semiring is antinegative if the only element having an additive inverse is the additive identity. For a commutative semiring S , a nonzero element s 2 S is called a zero divisor if there exists a nonzero element t 2 S such that s Á t ¼ 0.
Throughout this article, we let S be a commutative antinegative semiring containing the multiplicative identity with no zero divisors and Á is denoted by juxtaposition.
One of the simplest examples of an antinegative semiring without zero divisors is the binary Boolean algebra B which is a set of only two elements 0 and 1 with addition and multiplication on B defined as though 0 and 1 were integers, except that 1 þ 1 ¼ 1. Note that B is also called a Boolean semiring. Another example of these semirings is the fuzzy semiring which is the real interval ½0; 1 with the maximum and minimum as its addition and multiplication, respectively. Besides, any nonnegative subsemirings of R are other examples of such semirings.
Let M m;n ðS Þ denote the set of all m Â n matrices over S . The usual definitions for addition and multiplication of matrices are applied to such matrices as well. The notation M n ðS Þ is used when m ¼ n. A matrix in M m;n ðB Þ is called a Boolean matrix. A square matrix obtained by permuting rows of the identity matrix is called a permutation matrix.
In order to investigate LPPs on M m;n ðS Þ, we give the notions of a linear transformation on M m;n ðS Þ and when it preserves some certain properties.
Definition 2.2. A mapping T : M m;n ðS Þ ! M m;n ðS Þ is said to be a linear transformation if TðαX þ βYÞ ¼ αTðXÞ þ βTðYÞ for all X; Y 2 M m;n ðS Þ and α; β 2 S .
Let K be a subset of M m;n ðS Þ containing all matrices with a property P and T a linear transformation on M m;n ðS Þ. Then, we say that T preserves the property P if TðXÞ 2 K whenever X 2 K for all X 2 M m;n ðS Þ and T strongly preserves the property P if TðXÞ 2 K if and only if X 2 K for all X 2 M m;n ðS Þ.
We next state the most common concept of the standard form in the theory of linear preservers over semirings by the following definitions. Let N n ¼ 1; 2; . . . ; n f g .
Definition 2.3. A mapping σ : N n ! N n is said to be a half-mapping (on N n ) if σ satisfies the following properties: If a half-mapping is also a permutation on N n , then we call it a half-permutation.
Note that a half-mapping σ on N n is a permutation if n is even, but σ is not necessarily a permutation when n is odd. If n is odd, and the half-mapping σ on N n satisfies σ n 2 AE Ç À Á ¼ n 2 AE Ç , then σ is a permutation and hence a half-permutation.
Definition 2.4. Let e i be an n Â 1 Boolean matrix whose only nonzero entry is in the ith row, σ a half-mapping on N n . We define the matrix induced by σ to be the n Â n Boolean matrix whose the ith column is e σðiÞ if iÞ n 2 AE Ç and the n 2 AE Ç th column is e σ n 2 d e ð Þ when n is even, and e σ n 2 d e ð Þ þ e nÀσ n 2 d e ð Þþ1 when n is odd. The matrix induced by σ is denoted by P σ .
Definition 2.5. Let T : M m;n ðS Þ ! M m;n ðS Þ be a linear transformation. We say that T is induced by ðσ; τ; BÞ if there exist mappings σ and τ on N m and N n , respectively, and a matrix B 2 M m;n ðS Þ such that either (i) TðXÞ ¼ P σ ðX BÞQ τ for all X 2 M m;n ðS Þ; or (ii) m ¼ n and TðXÞ ¼ P σ ðX t BÞQ τ for all X 2 M m;n ðS Þ, where denotes the Schur product, i.e., A B ¼ ½a i;j b i;j for all A ¼ ½a i;j ; B ¼ ½b i;j 2 M m;n ðS Þ.
We let J m;n 2 M m;n ðS Þ denote the m Â n matrix whose entries equal to 1. From now on, the subscripts of matrices may be dropped when the sizes of matrices are clear from the context. We simply say that T is induced by ðσ; τÞ when B ¼ J. We also say that T is induced by σ if T is induced by ðσ; σ À1 Þ where σ is a permutation on N n . In general (see (Beasley & Pullman, 1987)), a linear transformation T on M m;n ðS Þ induced by ðσ; τ; BÞ where σ and τ are permutations on N m and N n , respectively, and all entries of B are nonzero, is usually called a ðP; Q; BÞ operator, where P and Q are permutation matrices induced by σ and τ, respectively.
The characterizations of linear transformations preserving term ranks were first studied by Beasley and Pullman (Beasley & Pullman, 1987) in 1987. This work is continued later on (see  and the references therein). We recall the definition of the term rank and its preserver theorem as follows.
Definition 2.6. The term rank of A 2 M m;n ðS Þ is the minimum number of rows or columns required to contain all the nonzero entries of A.
Theorem 2.7. (Beasley & Pullman, 1987) Let T : M m;n ðS Þ ! M m;n ðS Þ be a linear transformation. Then T preserves term ranks 1 and 2 if and only if T is induced by ðσ; τ; BÞ where σ and τ are permutations on N m and N n , respectively, and all entries of B are nonzero.
Let S ð0Þ n ðS Þ denote the set of all n Â n symmetric matrices with entries in S and all diagonal entries equal 0.
Let I be the n Â n identity matrix and K ¼ JnI. The following theorem is a characterization of linear transformations on S ð0Þ n ðB Þ preserving some term ranks provided by Beasley and his colleagues (Beasley et al., 2012). They also generalized this result to S (i) T preserves the star cover number 1 and TðKÞ ¼ K; (ii) T preserves the star cover numbers 1 and 2; (iii) T is induced by σ where σ is a permutation on N n .
To investigate further on bisymmetric matrices over semirings S , we let BS ð0Þ n ðS Þ denote the set of all n Â n bisymmetric matrices with entries in S and all diagonal and antidiagonal entries are 0.
n ðS Þ be the matrix whose the four entries at the ði; jÞ, ðj; iÞ, ðn À j þ 1; n À i þ 1Þ and ðn À i þ 1; n À j þ 1Þ positions are 1 and other entries are 0. The matrix Q i;j is called a quadrilateral cell and the matrix Q i;nÀjþ1 is called the corresponding quadrilateral cell of Q i;j . We let Ω n (or Ω when the size of matrices is clear) denote the set of all quadrilateral cells in BS ð0Þ n ðS Þ. For each i 2 N n 2 d e , the sum of all Q i;j such that j‚ i; n À i þ 1 f gis called the full double star on rows and The following notion is given in order to investigate more on the preserver problems of the matrix function on BS Definition 2.10. Let A 2 BS ð0Þ n ðS Þ. A double-star cover of A is a sum of full double stars that dominates A. The double-star cover number of A is the minimum number of full double stars whose sum dominates A.
Next, a generalization of a linear transformation on BS ð0Þ n ðS Þ induced by ðσ; σ À1 ; BÞ, which turns out to be the standard form for our results, is given by the following definition. We begin with the notation of certain square Boolean matrices.
n ðB Þ be a linear transformation. We say that T is induced by ðσ; F; GÞ if there exist a mapping σ on N n and matrices F ¼ ½f i;j ; G ¼ ½g i;j 2 M n ðB Þ such that n ðB Þ where Ω A is the set of all quadrilateral cells summing to A. We say that T is induced by ðσ; FÞ if T is induced by ðσ; F; GÞ and F ¼ G. Observe that To illustrate more clearly, we give the following example.
Example 2.12. Let σ be the half-mapping on N 7 defined by We end this section by introducing some instrumental notions.

Term rank preservers of bisymmetric matrices over Boolean semirings
In this section, we first consider linear preservers of bisymmetric matrices with zero diagonal and zero antidiagonal over Boolean semirings in order to generalize this notions to such matrices over other semirings. The following observations are obtained from the structure of the quadrilateral cells and the proofs are skipped.
(i) A is of term rank 2 or 4 if and only if A is a double-star matrix.
(ii) For each two distinct elements i; j 2 N n , A DS i and A DS j if and only if A Q i;j þ Q i;nÀjþ1 .
(iii) If the matrix L can be written as The study of linear transformations that strongly preserve a certain property is a common research focus (see (Beasley & Song, 2016a, 2016b and references therein). We provide a characterization of linear transformations on BS n ðB Þ preserving the matrix L strongly; i.e., L is the only matrix mapped to itself. Proof. The sufficient part is obvious. To show the necessary part, we first show that T is injective on Ω. Suppose on the contrary that there are two distinct quadrilateral cells Q u;v and Q w;z such that QÞ. This is a contradiction because Q u;v þ ∑ Q2Λ QÞL.
We suppose further that TðΩÞ 6 Ω. Since the images of the quadrilateral cells are not zero, it follows that there exists a quadrilateral cell Q p;q such that TðQ p;q Þ ¼ Q u 1 ;v 1 þ Á Á Á þ Q um;vm for some m ! 2 distinct quadrilateral cells. Note that m< Ω j j because T strongly preserves the matrix L. Let Δ denote the set Ωn Q u1;v1 ; . . . ; Q um;vm È É . Since TðLÞ ¼ L, for each Q i;j 2 Δ, there exists S i;j 2 Ω such that TðS i;j Þ ! Q i;j . Let Υ be the collection of such fixed S i;j 's. Thus γ j j Ω j j À m. Then This contradicts the fact that T strongly preserves L. Hence TðΩÞ Ω. Therefore, it follows by the finiteness of Ω that T is surjective on Ω.
Next, we prove one of the main results of this section. Proof. Note that T is bijective on Ω because of the properties of σ and F. Hence, the sufficient part follows.
To show the necessary part, we first show that for each nonzero matrix X 2 BS ð0Þ n ðB Þ, TðXÞ is not the zero matrix. Suppose on the contrary that there is a nonzero matrix X such that TðXÞ is zero.
Since X is nonzero, there is a quadrilateral cell Q X such that TðQÞ is zero. This is a contradiction because Q is a double-star matrix. Hence T is bijective on Ω by Lemma3.2.
First, we consider the case that n 5. There is nothing to do with the case n ¼ 1; 2 because BS ð0Þ 1 and BS ð0Þ 2 are the sets of the zero matrix.
Since T is bijective on Ω 3 , it follows that TðQ 1;2 Þ ¼ Q 1;2 , i.e., T is induced by ðσ; FÞ where σ is the identity map on N 3 and F is zero.
For n ¼ 5, we have Ω 5 ¼ Q 1;2 ; Q 1;3 ; Q 1;4 ; Q 2;3 È É . Since T preserves double-star matrices, and each of the three quadrilateral cells summing to DS 1 is mapped to three distinct quadrilateral cells, it follows that TðDS 1 Þ ¼ DS 1 or TðDS 1 Þ ¼ DS 2 . Similarly, TðDS 2 Þ ¼ DS 1 or TðDS 2 Þ ¼ DS 2 and we also obtain that TðDS 3 Þ ¼ DS 3 . This implies that T is induced by ðσ; FÞ where σ is the identity map on N 5 or σ ¼ ð12Þð3Þð45Þ and F is zero or F ¼ Q 1;2 þ Q 1;4 . Now, we consider the case that n ! 6. We next define a permutation σ on N n 2 d e . Since T preserves double-star matrices, TðDS i Þ is a double-star matrix for all 1 i To show that σ is well-defined, we suppose that there exist i 2 N n 2 d e such that TðDS i Þ DS p and TðDS i Þ DS q for some p; q 2 N n 2 d e with pÞq. Then, it follows from Lemma 3.1(ii) that TðDS i Þ Q p;q þ Q p;nÀqþ1 . Next, we calculate the numbers of quadrilateral cells that are summed to L and LnDS i . Note that ; if n is even: ( We observe that if n is odd and , then Ω DS i ¼ nÀ1 2 , otherwise Ω DS i ¼ n À 2. Since n ! 6, in both cases, the number of all quadrilateral cells that are summed to LnDS i is less than the number of quadrilateral cells excluding Q p;q and Q p;nÀqþ1 . Since L ¼ TðLÞ ¼ TðLnDS i Þ þ TðDS i Þ TðLnDS i Þ þ Q p;q þ Q p;nÀqþ1 , by the simple pigeonhole principle, the image of some quadrilateral cells dominates at least two quadrilateral cells. Then, there exists a quadrilateral cell Q x;y such that TðQ x;y Þ ! Q u;v þ Q w;z where u v, w z, u; w 2 1; 2; . . . ; n 2 Â Ã À 1 È É and ðu; vÞÞðw; zÞ. Since TðQ x;y Þ is a double-star matrix, there exists k 2 N n 2 ½ such that TðQ x;y Þ DS k . Hence, Q u;v Q u;v þ Q w;z TðQ x;y Þ DS k . Then, . For convenience, we may assume that v n 2 Â Ã . We then separate our proof into two cases and two subcases therein.
This means that DS u is the only double-star matrix that dominates Q u;v þ Q w;z . Since Q u;v þ Q w;z TðQ x;y Þ TðDS x Þ and TðDS x Þ is a double-star matrix, it follows that TðDS x Þ DS u . Similarly, TðDS y Þ DS u . Note that we can consider DS nÀyþ1 instead of DS y if y> n 2 AE Ç . Notice that the matrix L can be written as L ¼ DS x þ DS y þ DS z 1 þ Á Á Á þ DS z n 2 d e À3 for some z 1 ; . . . ; z n 2 d eÀ3 2 N n 2 d e nfx; yg. Then Thus, L is dominated by the sum of at most n 2 AE Ç À 2 full-star matrices. This is a contradiction.
If TðDS x Þ DS u and TðDS y Þ DS u , then we get a contradiction as the subcase 1.1. Similarly, TðDS x Þ DS u and TðDS y Þ DS u cannot occur simultaneously. This leaves us either (i) TðDS x Þ DS u and TðDS y Þ DS v or (ii) TðDS x Þ DS v and TðDS y Þ DS u . In any cases, TðQ x;y Þ Q u;v þ Q u;nÀvþ1 because Q x;y is dominated by both DS x and DS y and T is linear. Hence, In this case, we get a contradiction similarly to the subcase 1.1. Now we can conclude that σ is well-defined.
By Lemma, we have that T is bijective on Ω and since T is linear, T is bijective on BS n . This means that T maps fDS 1 ; . . . ; DS n 2 d e g onto fDS 1 ; . . . ; DS n 2 d e g injectively, and TðDS i Þ ¼ DS σðiÞ . Indeed, if n is odd, then TðDS n 2 d e Þ ¼ DS n 2 d e . Hence, σ is a permutation on N n 2 d e . Then, we extend σ to be a halfpermutation on N n . That is, σðiÞ ¼ n À σðn À i þ 1Þ þ 1 for all n 3 Â Ã i n.
We now construct an n Â n Boolean matrix F ¼ ½f i;j by letting if TðQ i;j Þ ¼ Q σðiÞ;σðjÞ 1; if TðQ i;j Þ ¼ Q σðiÞ;nÀσðjÞþ1 for all i; j 2 N n and iÞj.
Thus, for each A 2 BS ð0Þ n , TðAÞ ¼ ∑ That is T is induced by ðσ; FÞ, where σ is a halfpermutation and F is a tetrasymmetric matrix.
From Lemma 3.1(i), we can conclude that T preserves the set of all matrices of term ranks 2 and 4 if and only if T preserves double-star matrices. We observe that if T is induced by ðσ; FÞ, where σ is a half-permutation and F is a tetrasymmetric matrix, then T preserves term ranks 2 and 4. By these facts and the previous theorem, we obtain immediately the following results. We also provide the number of linear transformations satisfying such assumptions.
n ðB Þ preserving double-star cover number easily. From now on, we assume that n ! 5. if n is even; To show that σ is well-defined, suppose that there exists i 2 N n 2 d e such that TðDS i Þ DS p and TðDS i Þ DS q where p; q 2 N n 2 d e with pÞq. It follows from Lemma 3.1(ii) that TðDS i ÞQ p;q þ Q p;nÀqþ1 . Let j; k 2 N n 2 d e nfig with jÞk and j À i j j¼ 1. Then TðQ i;j þ Q j;k Þ ¼ TðQ i;j Þ þ TðQ j;k Þ TðDS i Þ þ TðQ j;k Þ Q p;q þ Q p;nÀqþ1 þTðQ j;k Þ. Since Q i;j þ Q j;k is of double-star cover number 1, it follows that TðQ j;k Þ DS p or TðQ j;k Þ DS q . We consider only the case that TðQ j;k Þ DS p because the other case is obtained similarly. Then TðDS i þ Q j;k Þ ¼ TðDS i Þ þ TðQ j;k Þ Q p;q þ Q p;nÀqþ1 þ DS p . This is a contradiction because DS i þ Q j;k is of double-star cover number 2, but Q p;q þ Q p;nÀqþ1 þ DS p is of double-star cover number 1. Hence σ is well-defined on N n 2 d e . Since T preserves double-star cover number 2, it implies that σ is a permutation on N n 2 d e . Next, we extend σ to be a half-mapping on N n . Now we consider the images of the quadrilateral cells. Let Q i;j 2 Ω. Without loss of generality, we may assume that i; j 2 N n 2 d e . Then, TðQ i;j Þ TðDS i Þ DS σðiÞ and TðQ i;j Þ TðDS j Þ DS σðjÞ . This implies that TðQ i;j Þ Q σðiÞ;σðjÞ þ Q σðiÞ;nÀσðjÞþ1 : Since TðQ i;j Þ is nonzero, Boolean matrices F ¼ ½f i;j and G ¼ ½g i;j can be constructed as follows. Let f i;i ¼ 0 ¼ f i;nÀiþ1 for all i 2 N n and n , it follows that Consequently, T is induced by ðσ; F; GÞ, where σ is a half-mapping on N n and F; G 2 BS ð0Þ n ðB Þ with F G.
Furthermore, if n is odd, then there exists s 2 N n 2 d e such that σðsÞ ¼ n 2 AE Ç . It follows that, for each tÞs, Q σðsÞ;σðtÞ ¼ Q σðsÞ;nÀσðtÞþ1 . That is f s;t ¼ 0 and g s;t ¼ 1 for all tÞs. Hence F< s G.
Next, we count the number of such linear transformations. We say that Λ i;j is free if ði; jÞ‚ [ tÞs Λ s;t . The following table shows the possible elements of F Λ i;j and G Λ i;j when Λ i;j is free.  We investigate further that if the condition`the linear transformation T preserves the matrix L' is assumed, then, in the proof of Theorem 3.5, the entries of matrices F and G are obtained as follows. For each i; j 2 N n 2 d e , if 1 2 F Λ i;j , then G Λ i;j ¼ 1 f g and G Λ i;j 0; 1 f g, otherwise. This leads us to the following corollary. The following lemma shows the relation between the term rank preservers and the doublestar cover number preservers. The proof is done by using Lemma 3.1(iii) and considering the image of each quadrilateral cell. The converse of Lemma 3.7 does not hold as the following example shows.

Term rank preservers of bisymmetric matrices over antinegative semirings
Let A ¼ ½a i;j 2 BS ð0Þ n ðS Þ. Beasley and Pullman (Beasley & Pullman, 1987) defined the pattern A of A to be the Boolean matrix whose ði; jÞth entry is 0 if and only if a i;j ¼ 0. We say that A has an L-pattern whenever A ¼ L. Note that the term ranks of A and A are equal and also the star cover numbers of A and A. For a linear transformation T on BS The following lemma is used to extend the results in BS ð0Þ n ðB Þ to BS ð0Þ n ðS Þ. Note that the proof of this lemma is straightforward so that it is left to the reader.
n ðS Þ be a linear transformation. We say that T is induced by ðσ; B; F; GÞ if there exist a mapping σ on N n , a matrix B 2 M n ðS Þ and matrices F ¼ ½f i;j ; G ¼ ½g i;j 2 M n ðB Þ such that TðAÞ ¼ ∑ n ðS Þ where Ω A is the set of all quadrilateral cells summing to A.
We say that T is induced by ðσ; B; FÞ if T is induced by ðσ; B; F; GÞ and F ¼ G.
n ðS Þ be a linear transformation. Then, T and T preserve the same term rank and the same double-star cover number.
The following characterizations are obtained by applying the results of the previous section and Lemma 4.2. Simply use the same methods of the proof of Corollary 3.3 in (Beasley et al., 2012) but with quadrilateral cells in place of digon cells. (i) T preserves double-star matrices and T strongly preserves L-pattern; (ii) T preserves term ranks 2 and 4, and T strongly preserves L-pattern; (iii) T is induced by ðσ; B; FÞ where σ is a half-permutation, F is a tetrasymmetric matrix and B 2 BS ð0Þ n ðSÞ is the matrix of L-pattern. (i) T preserves double-star matrices and T preserves L-pattern; (ii) T preserves the set of all matrices of term ranks 2 and 4, and T preserves L-pattern; (iii) T preserves double-star cover numbers 1 and 2, and T preserves L-pattern; (iv) T is induced by ðσ; B; F; GÞ where σ is a half-mapping on N n , F Λ G and F< s G provided n is odd and B 2 BS ð0Þ n ðS Þ is the matrix of L-pattern.

Conclusion
A new standard form is carefully given in order to characterize linear transformations preserving term ranks of bisymmetric matrices over semirings with some certain conditions. In this research, we investigate term rank preservers on bisymmetric Boolean matrices. The number of such linear transformations is also determined. Besides, the results on Boolean matrices are extended to matrices over semirings. Moreover, the characterizations of linear transformations preserving the special kind of the term rank, which is called the double-star cover number, on bisymmetric matrices over semirings are provided.
In our opinion, many questions can be further studied as shown in the following examples.
(1) Is it possible to drop the condition that all entries of diagonal and antidiagonal lines of bisymmetric matrices must be zero?
(2) Are there other characterizations of term rank preservers on bisymmetric matrices?
(3) What are characterizations of other rank preservers on bisymmetric matrices?