International Scholarly Research Notices
Volume 2015 (2015), Article ID 242515, 8 pages
http://dx.doi.org/10.1155/2015/242515

Research Article

Tan’s Epsilon-Determinant and Ranks of Matrices over Semirings

Preeti Mohindru and Rajesh Pereira

Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1

Received 28 November 2014; Accepted 9 January 2015

Academic Editor: Qing-Wen Wang

Copyright © 2015 Preeti Mohindru and Rajesh Pereira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We use the ϵ-determinant introduced by Ya-Jia Tan to define a family of ranks of matrices over certain semirings. We show that these ranks generalize some known rank functions over semirings such as the determinantal rank. We also show that this family of ranks satisfies the rank-sum and Sylvester inequalities. We classify all bijective linear maps which preserve these ranks.

1. Introduction

There are many equivalent ways of defining the rank of a matrix over a field. The rank of an by matrix could be defined as the largest for which there exists a by submatrix of with nonzero determinant, or the dimension of the row space of , or the dimension of the column space of or the smallest for which there exists an by matrix and a by matrix with . For matrices over semirings, all of these definitions are no longer equivalent and each of these generalizes to a distinct rank function for matrices over semirings. There are many different rank functions for matrices over semirings and their properties and the relationships between them have been much studied (see, e.g., [1–3]). In this paper, we use the -determinant of Tan [4, 5] to define a new family of rank functions for matrices over semirings. We examine the properties of these rank functions as well as their relationship to some of the other rank functions found in the literature.

In this section, we review some background material on semirings. In Section 2, we list the definition and some important properties of -determinant of Tan first introduced in [4, 5]. We then use the -determinant to introduce a new family of rank functions called the -rank functions and show that these rank functions satisfy some of the usual inequalities such as the rank-sum inequality and the Sylvester inequality. In Section 3, we look at bijective linear preservers of -rank for all matrices over commutative antinegative semirings. In Section 4, we introduce the sign pattern semiring and show that our rank function in this case is equal to the dimension of the largest sign-nonsingular submatrix. Our new rank functions depend on the ideal structure of the semirings and this leads us to study semirings which have a unique maximal ideal in Section 5. We show how the -rank generalizes determinantal rank in Section 6.

Semirings are a generalization of rings. Semirings satisfy all properties of unital rings except the existence of additive inverses. Vandiver introduced the concept of semiring in [6], in connection with the axiomatization of the arithmetic of the natural numbers.

Definition 1 (see [7]). A semiring is a set together with two operations and and two distinguished elements in with , such that

(1) is a commutative monoid,(2) is a monoid,(3) is both left and right distributive over ,(4)the additive identity 0 satisfies the property , for all .

In other words, semirings are unital rings without the requirement that each element has additive inverse. If is a commutative monoid then is called a commutative semiring. A semiring is said to be antinegative or zerosumfree if the only element with an additive inverse is the additive identity . An element of a semiring is called a unit if it has a multiplicative inverse in .

The natural numbers form a semiring under the usual addition and multiplication. A much studied example of a semiring is the max-plus semiring , where with and . In this case and [8]. A totally ordered set with greatest element 1 and least element 0 forms a semiring with and . This is called chain semiring. The chain semiring with two elements is called the Boolean semiring and it is denoted by .

Let denote the set of all by matrices over and denotes the set of all by matrices over . Addition and multiplication of these matrices can be defined in the usual way. Let be an by matrix. Then the element is called the -entry of . The -entry of is sometimes denoted by . Let and be by matrices over a semiring and let be an by matrix over the same semiring. Then and . The set of by matrices over a semiring is itself a semiring. For any , the matrix .

Definition 2 (see [9]). If is an by matrix over a commutative ring, then the standard determinant expression of is

where is the symmetric group of order and if is even permutation and if is odd permutation. Here is called a term of the determinant.

Since we do not have subtraction in a semiring, we cannot write the determinant of a matrix over a semiring in this form. We split the determinant into two parts, the positive determinant and the negative determinant.

Definition 3 (see [9]). Let be an by matrix over a commutative semiring . We define the positive and the negative determinant as

Here is the alternating group of order , that is, the set of all even permutations of order and is the set of all odd permutations of order .

As such we note that the determinant of a matrix over a commutative ring takes the form

In the semiring case, one cannot subtract the negative determinant from the positive determinant and so the positive determinant and the negative determinant are listed as a pair. This pair is called the bideterminant.

Definition 4 (see [10]). Let be an by matrix over a commutative semiring. The bideterminant of is .

The definition of the permanent involves no subtractions; hence it carries over to the semiring case unchanged.

Definition 5 (see [9]). Let be an by matrix over a semiring; then the permanent of is

The permanent of a square matrix is the sum of its positive and negative determinants: .

Finally we note that we have a canonical preorder (a reflexive transitive relation) called the difference preorder on semirings.

Definition 6. Let be a semiring. We define the difference preorder on as follows: if then if there exists such that .

The difference preorder may not be a partial order. However for many semirings such as the nonnegative semiring, max-plus semiring, and any Boolean algebra or distributive lattice, the difference semiring corresponds to the natural order on the set.

2. The -Determinant and -Rank

In [4, 5], Tan introduced a new type of determinant for semirings. We begin with a concept formulated independently by Akian et al. in [1] and Tan in [4].

Definition 7 (see [1, 4, 5]). Let be a semiring. A bijection is called a symmetry if for all and for all .

The term symmetry is from [1]; this similar concept is called an -function in [4, 5]. We note that all of these references also required a symmetry to be additive (i.e., ). In [1], a symmetry must further satisfy . We have removed these conditions from the definition as we will show they follow from other properties of a symmetry. One can easily characterize all symmetries in a semiring.

Proposition 8. Let be a commutative semiring. A function is a symmetry if and only if there exists an such that and for all .

Proof. Suppose is a symmetry. Let ; then . Furthermore . The other direction is a straightforward verification.

It follows easily from the previous proposition that if is a symmetry and then , , and .

We use this observation to slightly restate the definition of an -determinant given in [4, 5]. We will use to denote the element whose square is the identity rather than a symmetry or -function as in [4, 5]; this allows us to use the same terminology and notation as in [4, 5] while taking advantage of the characterization of symmetries given in Proposition 8.

Definition 9. Let be a commutative semiring and let satisfy . Then is defined as the following function: .

We will get a distinct symmetry on and hence a distinct from every choice of which satisfies . One candidate for that exists in every semiring is the multiplicative identity . If , we get .

Tan has shown that the -determinant satisfies two very important identities analogous to the Binet-Cauchy theorem and the Laplace expansion of the ordinary determinant. In order to state them, we introduce the following notation. Let and be two subsets of of equal cardinality. Let denote the by submatrix of whose th entry is . We define .

Theorem 10 (see [5, Theorem 3.5] (the generalized Binet-Cauchy theorem)). Let be a commutative semiring and let satisfy . Let , , . Let and be two subsets of of cardinality . Then there exists a such that .

Theorem 11 (see [5, Theorem 3.3] (the generalized Laplace expansion)). Let be a commutative semiring and let satisfy . If and , then .

In the case where is a commutative ring and , the -determinant reduces to the regular determinant and the two theorems above reduce to the usual Binet-Cauchy theorem and the Laplace expansion.

One corollary of the generalized Binet-Cauchy theorem is the following difference preorder inequality for square matrices.

Corollary 12. Let be a commutative semiring and let satisfy . The inequality holds for all .

The special case of this result where is a Boolean algebra (and by necessity which means is the permanent) has appeared in [11].

We can use Tan’s generalization of the Laplace expansion to obtain a relationship between the -determinant of and those of various submatrices of and .

Corollary 13. Let , let be a commutative semiring, and let satisfy . The identity

holds for all .

Proof. Let be the by matrix whose th row is equal to the th row of if and whose th row is equal to the th row of if . Then using the multilinearity of the -determinant, . Now we use the Laplace expansion on , for any fixed , to get

Hence

Definition 14. Let be a commutative semiring and let be an by matrix over . If , let be the ideal in generated by the set of all the by -minors of . One defines and when .

It follows immediately from the Laplace expansion that for all .

Definition 15. Let be a commutative semiring and let be an element of such that and that is not a unit. Let be any proper ideal of which contains . The -rank of an by matrix (denoted by ) is the largest nonnegative integer such that is not contained in .

For certain semirings, it may be possible that there is no choice of for which is not a unit. An example of this is the semiring of nonnegative real numbers with the usual addition and multiplication. In this case the only satisfying is itself and is a unit. The max-plus and max-min semirings are other examples of this. For semirings such as these, one cannot immediately define an -rank. We will examine how to handle cases like this in the last section of the paper.

The principal ideal generated by is a natural choice for our ideal .

Definition 16. Let be a commutative semiring and let be an element of such that and that is not a unit. Let be the principal ideal generated by . The -rank of an by matrix (denoted by ) is the largest nonnegative integer such that is not contained in .

It is clear that any containing contains and therefore . Recall that the standard definition of the rank of a matrix over a ring is the size of the largest for which the ideal generated by all by subdeterminants of the matrix is nonzero [12, page 82]. If is a ring, then the -rank of a matrix over is equal to the standard ring-theoretic rank of the matrix over the quotient ring where is the natural entrywise quotient map.

We now examine some inequalities satisfied by these ranks that are implied by the Binet-Cauchy theorem, the Laplace expansion, and our determinant sum identity. The first is the relationship between this rank and the factor rank. We begin by reminding readers of the definition of the factor rank.

Definition 17. Let be a commutative semiring and . The factor rank (or Schein rank) of is the smallest integer for which there exists an by matrix and an by matrix such that . The factor rank is denoted by .

Proposition 18. The inequality holds whenever is a commutative semiring and .

Proof. Let , then there exist matrices and such that . Let be the matrix obtained by adding a zero column to the right of and let be the matrix obtained by adding a zero row to the bottom of . Clearly . Now we compute the -minors of of order , using the Binet-Cauchy theorem; that is, , . Since the first summand is , so is contained in the ideal generated by . Hence .

We can also prove a version of Sylvester’s inequality for the -rank.

Proposition 19. Let be a commutative semiring and let be an element of such that and that is not a unit. Let be an proper ideal of which contains . The inequality holds for all and .

Proof. Let . If we are done so suppose and let and be both arbitrary subsets of of cardinality . Then either or is contained in . It follows from the Binet-Cauchy theorem that .

It should be noted that the condition is required for our version of Sylvester’s inequality to hold; this is largely our motivation for insisting on this condition.

We also have the following rank-sum inequality.

Proposition 20. Let be a commutative semiring and let be an element of such that and that is not a unit. Let be an proper ideal of which contains . The inequality holds for all .

Proof. We begin by proving the inequality in the special case where and . Hence . Let us suppose that . This implies that . We can use Corollary 13 to show that . Note that every term in the expansion of is of a power of times where and are subsets of satisfying . Let . If , then and since we must have . Similarly, if , then . Therefore every term in the expansion of is in and hence .
Now we prove the general case. Let . If then we are done so suppose . Now let and be subsets of and , respectively, both of cardinality . Then . Hence and since is an arbitrary by submatrix of , we have .

3. Bijective Linear -Rank Preservers

In this section, we look at bijective linear operators which preserve -rank of matrices over antinegative commutative semiring.

Definition 21 (see [2]). Let be a commutative semiring and be an by matrix over . The term rank of is the minimum number of lines rows and columns needed to include all nonzero entries of . The term rank of a matrix is denoted by .

Proposition 22 (see [2]). For any commutative semiring, one has whenever is an by matrix over .

Let be a semiring and . We write if there exists such that . We note that the relation is a reflexive and transitive relation but not antisymmetric in general. Therefore it is a preorder. It is easy to check that any linear operator preserves this preorder. Further, if is an antinegative semiring then the term rank is a monotone function; that is, if then .

Definition 23. Let be a commutative semiring and . The Schur product of and , denoted as , is an by matrix whose th entry is .

Definition 24. Let be a commutative semiring. A matrix is called a submonomial matrix if every line row or column of contains at most one nonzero entry. A matrix is called a monomial matrix if every line row or column of contains exactly one nonzero entry.

The concept of operator is a fundamental concept in the theory of linear preservers over semirings.

Definition 25 (see [13]). Let be a linear operator from to itself. One says that is a strong operator if there exist , , and such that and are permutation matrices, and all of the entries of are units and either or and .

We also use a theorem from the same reference. We note though there is an error earlier in [13] for the definition of the term rank, the following theorem is correct as stated as it only uses correct properties of the term rank.

Theorem 26 (see [13, Theorem 2.12]). Let be a commutative antinegative semiring. If is a function which satisfies for all with equality whenever is a submonomial matrix, then any bijective linear operator which preserves this rank function must be a strong operator.

Since the -rank satisfies the hypotheses of the above theorem, we now have the following corollary which classifies all bijective linear operators which preserve the -rank.

Corollary 27. Let be a commutative antinegative semiring and let be an element of such that and that is not a unit. Let be any proper ideal of which contains . Any bijective -rank preserver on must be a strong operator.

4. Sign Pattern Matrices and the Sign Pattern Semiring

In this section, we explore connections between the sign pattern matrices and -rank.

A matrix whose entries are from the set is called a sign pattern matrix. If is a real matrix, then the sign pattern of is obtained from , by replacing each entry by its signs [14, 15]. The sign pattern of is denoted by , where

Thus in a sign pattern matrix all we know is the sign of each entry. We do not know the exact values of the entries. We denote the set of all by sign pattern matrices by . Sometimes we may not know the signs of certain entries, so a new symbol, , has been introduced to denote such entries.

Definition 28 (see [16]). The generalized sign pattern matrices are the matrices over the set , where corresponds to entries where the sign is unknown.

The set can be viewed as a semiring. If , then is a commutative semiring with identity, where the operations of addition and multiplication are defined as follows:

Clearly all the properties of a semiring are satisfied where is the additive identity and is multiplicative identity. Here and are the units of . More about the sign pattern semiring can be found in [17].

Definition 29 (see [14]). Let be a real matrix. The qualitative class of is , the set of all real matrices with the same sign pattern as .

Definition 30 (see [14]). A sign pattern matrix is called sign-nonsingular (SNS) if every matrix in its qualitative class is nonsingular.

For matrices over the sign pattern semiring, we can give a more concrete interpretation of the -rank. The sign pattern semiring has only two elements whose square is the identity, namely, and . The ideal generated by is the entire semiring but generates the unique proper ideal . Therefore is the only available choice for and we have a unique -rank. Hence . It is easy to show that an by sign pattern matrix has -rank if and only if it is an SNS matrix. Hence the -rank of a sign pattern matrix is the largest integer for which there exists a by SNS submatrix of .

5. Sublocal Semirings

It was remarked earlier that . In other words amongst the family of -ranks, choosing to be the ideal generated by gives us the largest possible rank function from this family. The minimal rank functions from this family arise from the choice of to be a maximal ideal which contains . In general, there may be many maximal ideals. In this section, we will look at semirings which have a unique maximal ideal. We will use the term sublocal semiring to denote a semiring which has a unique maximal ideal. Sublocality in semirings is essentially the straightforward generalization of the very useful concept of locality in rings. We use the term sublocal semiring because the term local semiring has been used to define a slightly different concept.

Definition 31 (see [18]). An ideal of a commutative semiring is called a -ideal if for any with , one has .

We note that if we consider a commutative ring to be semiring, the semiring ideals of are exactly the semiring -ideals of which are also exactly the ring ideals of .

Definition 32 (see [18]). A proper ideal of a commutative semiring is called a maximal resp., -maximal ideal if there exists no other proper ideal resp., -ideal such that .

Definition 33 (see [18]). Let be a commutative semiring. One says that is a local semiring if has only one -maximal ideal.

Now we will define sublocal semirings using maximal ideals instead of -maximal ideals. This is useful as some semirings do not have proper -ideals. For example, the sign pattern semiring has only one proper ideal and this is not a -ideal.

Definition 34. Let be a commutative semiring. One says that is a sublocal semiring if has only one maximal ideal.

We note that both local and sublocal semirings are direct but different semiring generalizations of the concept of a local ring. Local semirings have been useful in semiring theory; see [18] for examples of this. We will show that sublocality is a useful property as well. There are many examples of sublocal semirings; we list some of the notable ones. The sign pattern semiring is a sublocal semiring having only one maximal ideal . We note that this maximal ideal is contained in the proper subsemiring ; however fails to be an ideal in the sign pattern semiring. The set of all natural numbers, , forms a sublocal semiring whose only one maximal ideal . All chain semirings are sublocal semirings with as a unique maximal ideal. A semifield is a commutative semiring in which all elements except have a multiplicative inverse. (The Boolean and max-plus semirings are examples of semifields.) All semifields are sublocal semirings as the zero ideal is the unique maximal ideal.

We begin with the following elementary lemma whose proof is identical to the corresponding result for rings.

Lemma 35 (see [18]). An element of a commutative semiring is a unit of if and only if lies outside all maximal ideals of .

Proof. Let be a unit of ; then the ideal generated by must be itself and hence lies outside all maximal ideals of . If is not a unit of , then and hence there exits a maximal ideal of such that .

One of the key results of [18] is that a semiring is local if the set of all of its nonunits forms a -ideal. The analog for sublocal semirings is an easy consequence of Lemma 35.

Corollary 36. A commutative semiring is a sublocal semiring if and only if the set of all nonunits of forms an ideal.

Since every -ideal is an ideal, it follows that every local semiring is a sublocal semiring. The converse is false. Consider the nonnegative integers with the usual addition and multiplication. The unique maximal ideal is ; the maximal -ideals are of the form for any prime .

Since the set of nonunits in any sublocal semiring is an ideal, the nonunits are closed under addition. Hence we have the following observation which will prove useful later on.

Corollary 37. Let be a sublocal semiring. Let . If is a unit of , then either or is a unit of .

6. Symmetrized Semirings

The ranks introduced in the previous sections all require an element satisfying the condition that and is not a unit. Such an element may not exist in a given semiring; the max-min and max-plus semirings are examples of semiring which lack an . Fortunately, there is a known construction which allows us to append such an element. This construction is from [8], in which it was applied to the max-plus semiring. In this paper we explore applications of this construction both to general semirings and to the specific examples such as the Boolean semiring and the sign pattern semiring.

If is a commutative semiring then is also a commutative semiring with addition and multiplication defined as follows: for all and ,

We can see that all the properties of a semiring are satisfied with being the additive identity of . Essentially this construction allows us to append an element with the property to the semiring in a natural way giving us a way to apply the -determinant theory to semirings which do not have nontrivial self inversive elements. The ideal in generated by is which we will call the diagonal ideal. The -determinant in this case is the standard bideterminant and the -rank is the standard determinantal rank defined as follows.

Definition 38. Let be an by matrix over a commutative semiring . The determinantal rank of is the largest for which there exists , a by submatrix of with .

The determinantal rank has been much studied (see [1], for instance). Our results in Section two generalize the known results on the determinantal rank of max-plus matrices to the -rank of matrices over general semirings.

Remark 39. Recall that denotes Boolean semiring which has two elements . Then = is also a semiring with the addition and the multiplication defined above for . Moreover it is isomorphic to the sign pattern semiring as , , , and .

We complete our paper by showing that the symmetrized semiring inherits some important properties from .

Theorem 40. Let be a commutative semiring. If is antinegative and has no zero divisors then is also antinegative and has no zero divisors.

Proof. Since is antinegative, only has an additive inverse. Let us suppose that has an additive inverse, so there exists , such that . Consequently and . Since is antinegative so . Hence . Thus only the additive identity has an additive inverse in which means is an antinegative semiring. Now suppose that is a zero divisor, so there exists a nonzero , such that . It follows that and . Since is antinegative, and . Also has no zero divisors so either or and either or ; combining this with , we get either or . Hence has no zero divisors.

If is a semiring, we let denote the set of units of . There is a very close relation between the units of and the units of .

Lemma 41. If S is a commutative antinegative semiring with no zero divisors then .

Proof. Let be a unit in . Then there exists a nonzero element in such that . Consider and . Consequently and are units of . Now we have to prove that these are the only units for . Suppose is a unit in . Then there exists a nonzero element such that . Thus . Consequently and . Since and is an antinegative semiring, so and . Also given that has no zero divisors it follows that either or (note that both and cannot be zero because ( and either or (here also both and cannot be zero because . Since and are nonzero elements of so the units of , , have only two choices which are and . Putting in , we get , and this means that is a unit of . Putting in (, we get , and this means that is a unit of . Thus all the units in are of the type and where is a unit in .

We can now prove that if is a sublocal antinegative semiring with no zero divisors then so is .

Theorem 42. If is a sublocal antinegative semiring with no zero divisors then is also a sublocal antinegative semiring with no zero divisors.

Proof. Suppose is a sublocal antinegative semiring with no zero divisors; then by Theorem 40, is an antinegative semiring with no zero divisors. Hence we only need to prove that is sublocal; we show this by proving that the set of all nonunits in forms an ideal of . Let be the set of all nonunits of . Let and such that , where is a unit in . Hence and . Since is antinegative so and , and also , where is a unit in so (using Corollary 37) either is a unit or is a unit. Thus either , where is a unit in , or , where is a unit in . It follows that either is a unit or is a unit in , which is a contradiction to the fact that both and . Thus the sum of nonunits in is a nonunit. A similar argument works if , where is a unit in . Now suppose that for and we have , where is a unit in . Then and . Since is antinegative so and , and also has no zero divisors so either or and either or . Clearly or cannot be since , so we have or . Further is a unit in and , so either is a unit in or is a unit in . Hence either is a unit in or is a unit in . We get or is a unit in , which is a contradiction to the fact that . Thus for all and . Hence the set of all nonunits in forms an ideal of . Consequently is a sublocal semiring. A similar argument works if , where is a unit in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author would like to acknowledge the support of the Province of Ontario in the form of a Queen Elizabeth II Graduate Scholarship in Science and Technology (QEII-GSST). The second author would like to acknowledge the support of an NSERC Discovery Grant no. 400550. Both authors would like to thank the referee for many suggestions which significantly improved this paper.

References

M. Akian, S. Gaubert, and A. Guterman, “Linear independence over tropical semirings and beyond,” in Tropical and Idempotent Mathematics, G. L. Litvinov and S. N. Sergeev, Eds., vol. 495 of Contemporary Mathematics, pp. 1–38, American Mathematical Society, 2009.

L. B. Beasley and A. E. Guterman, “Rank inequalities over semirings,” Journal of the Korean Mathematical Society, vol. 42, no. 2, pp. 223–241, 2005.

L. B. Beasley and N. J. Pullman, “Semiring rank versus column rank,” Linear Algebra and Its Applications, vol. 101, pp. 33–48, 1988.

Y. J. Tan, “On invertible matrices over commutative semirings,” Linear and Multilinear Algebra, vol. 61, no. 6, pp. 710–724, 2013.

Y.-J. Tan, “Determinants of matrices over semirings,” Linear and Multilinear Algebra, vol. 62, no. 4, pp. 498–517, 2014.

H. S. Vandiver, “Note on a simple type of algebra in which the cancellation law of addition does not hold,” Bulletin of the American Mathematical Society, vol. 40, no. 12, pp. 914–920, 1934.

J. S. Golan, “Semirings for the ring theorist,” Revue Roumaine de Mathématique Pures et Appliquées, vol. 35, no. 6, pp. 531–540, 1990.

M. Akian, G. Cohen, S. Gaubert, R. Nikoukhah, and J. P. Quadrat, “Linear systems in (max, +) algebra,” in Proceedings of the 29th Conference on Decision and Control, pp. 151–156, IEEE, Honolulu, Hawaii, USA, December 1990.

P. L. Poplin and R. E. Hartwig, “Determinantal identities over commutative semirings,” Linear Algebra and Its Applications, vol. 387, pp. 99–132, 2004.

M. Gondran and M. Minoux, “L 'indépendance linéaire dans les dioïdes,” Électricité de France. Bulletin de la Direction des Études et Recherches. Série C. Mathématiques. Informatique, no. 1, pp. 67–90, 1978.

J. M. Simões-Pereira, “Boolean permanents, permutation graphs and products,” SIAM Journal on Applied Mathematics, vol. 16, no. 6, pp. 1251–1254, 1968.

W. C. Brown, Matrices Over Commutative Rings, Marcel Dekker, New York, NY, USA, 1993.

R. Pereira, “Bijective linear rank preservers for spaces of matrices over antinegative semirings,” Linear Algebra and Its Applications, vol. 435, no. 7, pp. 1666–1671, 2011.

R. A. Brualdi and B. L. Shader, Matrices of Sign-Solvable Linear Systems, vol. 116 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1995.

F. J. Hall, Z. Li, and B. Rao, “From Boolean to sign pattern matrices,” Linear Algebra and Its Applications, vol. 393, pp. 233–251, 2004.

L. Zhongshan, H. Frank, and C. Eschenbach, “On the period and base of a sign pattern matrix,” Linear Algebra and Its Applications, vol. 212-213, pp. 101–120, 1994.

Yu. A. Al'pin and S. N. Il'in, “Powers of sign portraits of real matrices,” Journal of Mathematical Sciences, vol. 121, no. 4, pp. 2441–2447, 2004.

R. E. Atani and S. E. Atani, “Ideal theory in commutative semirings,” Buletinul Academiei de Ştiinţe a Republicii Moldova, Matematica, no. 257, pp. 14–23, 2008.