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Tan's epsilon-determinant and ranks of matrices over semirings.

1. Introduction

There are many equivalent ways of defining the rank of a matrix over a field. The rank of an m by n matrix A could be defined as the largest k for which there exists a k by k submatrix of A with nonzero determinant, or the dimension of the row space of A, or the dimension of the column space of A or the smallest k for which there exists an m by k matrix B and a k by n matrix C with A = BC. 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 [epsilon]-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 [epsilon]-determinant of Tan first introduced in [4, 5]. We then use the [epsilon]-determinant to introduce a new family of rank functions called the ([epsilon], I)-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 ([epsilon], I)-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 ([epsilon], I)-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 S together with two operations [direct sum] and [cross product] and two distinguished elements 0,1 in S with 0 [not equal to] 1, such that

(1) (S, [direct sum] 0) is a commutative monoid,

(2) (S, [cross product] 1) is a monoid,

(3) [cross product] is both left and right distributive over [direct sum],

(4) the additive identity 0 satisfies the property r [cross product] 0 = 0 [cross product] r = 0, for all r [member of] S.

In other words, semirings are unital rings without the requirement that each element has additive inverse. If (S, [cross product], 1) is a commutative monoid then S 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 0. An element of a semiring S is called a unit if it has a multiplicative inverse in S.

The natural numbers form a semiring under the usual addition and multiplication. A much studied example of a semiring is the max-plus semiring ([R.sub.max], [direct sum], [cross product]), where [R.sub.max] = R [universal] {-[infinity] with a [direct sum] b = max{a, b} and a [cross product] b = a + b. In this case 0 = -[infinity] and 1 = 0 [8]. A totally ordered set S with greatest element 1 and least element 0 forms a semiring with a [direct sum] b = max{a, b} and a [cross product] b = min{a, b}. This is called chain semiring. The chain semiringwith two elements {0,1} is called the Boolean semiring and it is denoted by [beta].

Let [M.sub.mn](S) denote the set of all m by n matrices over S and [M.sub.n](S) denotes the set of all n by n matrices over S. Addition and multiplication of these matrices can be defined in the usual way. Let A be an m by n matrix. Then the element [a.sub.ij] is called the (i, j)-entry of A. The (i, j)-entry of A is sometimes denoted by [A.sub.ij]. Let A = [[a.sub.ij]] and B = [[b.sub.ij]] be m by n matrices over a semiring (S, [direct sum], [cross product]) and let C = [[c.sub.ij]] be an n by p matrix over the same semiring. Then A + B = [[a.sub.ij] [direct sum] [b.sub.ij]] and AC = [[[direct sum].sup.n.sub.k=1][a.sub.ik] [cross product][c.sub.kj]]. The set of n by n matrices over a semiring is itself a semiring. For any k [member of] S, the matrix kA = [k [cross product] [a.sub.ij]].

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [S.sub.n] is the symmetric group of order n and sgn([sigma]) = + 1 if [sigma] is even permutation and sgn([sigma]) = -1 if [sigma] is odd permutation. Here sgn([sigma])[a.sub.1[sigma](1)] [a.sub.2[sigma](2)] ... [a.sub.n[sigma](n)] 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 A be an n by n matrix over a commutative semiring S. We define the positive and the negative determinant as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Here [A.sub.n] is the alternating group of order n, that is, the set of all even permutations of order n and [S.sub.n]\[A.sub.n] is the set of all odd permutations of order n.

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

det (A) = [det.sub.+] (A) - [det.sub._] (A). (3)

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 A be an n by n matrix over a commutative semiring. The bideterminant of A is ([det.sub.+](A), [det.sub.-](A)).

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

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

The permanent of a square matrix is the sum of its positive and negative determinants: per(A) = [det.sub.+](A) [direct sum] [det.sub.-](A).

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

Definition 6. Let S be a semiring. We define the difference preorder [greater than or equal to] on S as follows: if x, y [member of] S then x [greater than or equal to] y if there exists z [member of] S such that x = y [direct sum] z.

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 [epsilon]-Determinant and [epsilon]-Rank

In [4,5], Tan introduced a newtype 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 S be a semiring. A bijection [tau] : S [right arrow] S is called a symmetry if [tau]([tau](a)) = a for all a [member of] S and [tau](a [cross product] b) = a [cross product] [tau](b) = [tau](a) [cross product] b for all a,b [member of] S.

The term symmetry is from [1]; this similar concept is called an [epsilon]-function in [4, 5]. We note that all of these references also required a symmetry to be additive (i.e., [tau](a [direct sum] b) = [tau](a) [direct sum] [tau](b)). In [1], a symmetry [tau] must further satisfy [tau](0) = 0. 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 S be a commutative semiring. A function [tau] : S [right arrow] S is a symmetry if and only if there exists an [epsilon] [member of] S such that [[epsilon].sup.2] = 1 and [tau](x) = [epsilon][cross product] x for all x [member of] S.

Proof. Suppose [tau] : S [right arrow] S is a symmetry. Let [epsilon] = [tau](1); then [tau](x) = [tau](1 [cross product] x) = [epsilon] [cross product] x. Furthermore [[epsilon].sup.2] = [epsilon] [cross product][tau](1) = [tau]([epsilon] [cross product] 1) = 1. The other direction is a straightforward verification.

It follows easily from the previous proposition that if [tau] : S [right arrow] S is a symmetry and x, y [member of] S then [tau](x [direct sum] y) = [tau](x) [direct sum] [tau](y), [tau](0) = 0, and [tau](x) [cross product] [tau[(y) = x [cross product] y.

We use this observation to slightly restate the definition of an [epsilon]-determinant given in [4, 5]. We will use [epsilon] to denote the element whose square is the identity rather than a symmetry or [epsilon]-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 S be a commutative semiring and let [epsilon] [member of] S satisfy [[epsilon].sup.2] = 1. Then [det.sub.[epsilon]] : [M.sub.n](S) [right arrow] S is defined as the following function: [det.sub.[epsilon]] (A) = [det.sub.+](A) [direct sum] ([epsilon] [cross product] det_(A)).

We will get a distinct symmetry on S and hence a distinct [det.sub.[epsilon]] from every choice of [epsilon] [member of] S which satisfies [[epsilon].sup.2] = 1. One candidate for e that exists in every semiring is the multiplicative identity 1. If [epsilon] = 1, we get [det.sub.[epsilon]] (A) = per(A).

Tan has shown that the [epsilon]-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 [alpha] = {[[alpha].sub.1] < [[alpha].sub.2] < ... < [[alpha].sub.k]} and [beta] = {[[beta].sub.1] < [[beta].sub.2] < ... < [[beta].sub.k]} be two subsets of {1,2, ..., n} of equal cardinality. Let A[[alpha] | [beta]] denote the k by k submatrix of A whose (i, j)th entry is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We define [pi]([alpha]) = [[summation].sup.k.sub.j=1] [[alpha].sub.j].

Theorem 10 (see [5, Theorem 3.5] (the generalized Binet-Cauchy theorem)). Let S be a commutative semiring and let [epsilon] [member of] S satisfy [[epsilon].sup.2] = 1. Let A [member of] [M.sub.m,n]), B [member of] [M.sub.n,p](S), 1 [less than or equal to] k [less than or equal to] min(m,n, p). Let [alpha] and [beta] be two subsets of {1,2, ..., n} of cardinality k. Then there exists a [delta] [member of] S such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 11 (see [5, Theorem 3.3] (the generalized Laplace expansion)). Let S be a commutative semiring and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the case where S is a commutative ring and [epsilon] = -1, the [epsilon]-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 S be a commutative semiring and let [epsilon] [member of] S satisfy [[epsilon].sup.2] = 1. The inequality [det.sub.[epsilon]](AB) [greater than or equal to] [det.sub.[epsilon]](A) [cross product] [det.sub.[epsilon]] (B) holds for all A, B [member of] [M.sub.n](S).

The special case of this result where S is a Boolean algebra (and by necessity [epsilon] = 1 which means [det.sub.[epsilon]] is the permanent) has appeared in [11].

We can use Tan's generalization of the Laplace expansion to obtain a relationship between the [epsilon]-determinant of A + B and those of various submatrices of A and B.

Corollary 13. Let n [member of] N, let S be a commutative semiring, and let [epsilon] [member of] S satisfy [[epsilon].sup.2] = 1. The identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

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

Proof. Let [C.sub.[alpha]] be the n by n matrix whose ith row is equal to the ith row of A if i [member of] [alpha] and whose ith row is equal to the th row of B if i [not member of] [alpha]. Then using the multilinearity of the [epsilon]-determinant, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now we use the Laplace expansion on [det.sub.[epsilon]] ([C.sub.[alpha]]), for any fixed [alpha], to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Definition 14. Let S be a commutative semiring and let A be an m by n matrix over S. If 1 [less than or equal to] k [less than or equal to] min(m, n), let [I.sup.[epsilon].sub.k](A) be the ideal in S generated by the set of all the k by k [epsilon]-minors of A. One defines [I.sup.[epsilon].sub.0](A) = S and [I.sup.[epsilon].sub.k] (A) = {0} when k > min(m, n).

It follows immediately from the Laplace expansion that [I.sup.[epsilon].sub.k](A) [subset or equal to][I.sup.[epsilon].sub.k-1](A) for all k [member of] N.

Definition 15. Let S be a commutative semiring and let [epsilon] be an element of S such that [[epsilon].sup.2] = 1 and that 1 [direct sum] [epsilon] is not a unit. Let I be any proper ideal of S which contains 1 [direct sum] [epsilon]. The ([epsilon], I)-rank of an m by n matrix A (denoted by [rank.sup.[epsilon],I.sub.det](A)) is the largest nonnegative integer k such that [I.sup.[epsilon].sub.k](A) is not contained in I.

For certain semirings, it may be possible that there is no choice of [epsilon] for which 1 [direct sum] [epsilon] is not a unit. An example of this is the semiring of nonnegative real numbers [R.sup.+] with the usual addition and multiplication. In this case the only [epsilon] [member of] [R.sup.+] satisfying [[epsilon].sup.2] = 1 is 1 itself and 2 = 1 + 1 is a unit. The maxplus and max-min semirings are other examples of this. For semirings such as these, one cannot immediately define an ([epsilon], I)-rank. We will examine how to handle cases like this in the last section of the paper.

The principal ideal generated by 1 [direct sum] [epsilon] is a natural choice for our ideal I.

Definition 16. Let S be a commutative semiring and let [epsilon] be an element of S such that [[epsilon].sup.2] = 1 and that 1 [direct sum] [epsilon] is not a unit. Let [I.sub.1[direct sum][epsilon]] be the principal ideal generated by 1 [direct sum] [epsilon]. The [epsilon]-rank of an m by n matrix A (denoted by [rank.sup.[epsilon].sub.det](A)) is the largest nonnegative integer k such that [I.sup.[epsilon].sub.k](A) is not contained in [I.sub.1[direct sum][epsilon]].

It is clear that any I containing 1 [direct sum] [epsilon] contains [I.sub.1[direct sum][epsilon]] and therefore [rank.sup.[epsilon],I.sub.det](A) [less than or equal to] [rank.sup.[epsilon].sub.det] (A). Recall that the standard definition of the rank of a matrix over a ring is the size of the largest k for which the ideal generated by all k by k subdeterminants of the matrix is nonzero [12, page 82]. If S is a ring, then the ([epsilon], I)-rank of a matrix A over S is equal to the standard ring-theoretic rank of the matrix [phi](A) over the quotient ring S/I where [phi] 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 S be a commutative semiring and A e [M.sub.m,n](S). The factor rank (or Schein rank) of A is the smallest integer r for which there exists an m by r matrix B and an r by n matrix C such that A = BC. The factor rank is denoted by f(A).

Proposition 18. The inequality [rank.sup.[epsilon].sub.det](A) [less than or equal to] f(A) holds whenever S is a commutative semiring and A [member of] [M.sub.m,n](S).

Proof. Let r = f(A), then there exist matrices B [member of] [M.sub.m,r](S) and C [member of] [M.sub.r,n](S) such that A = BC. Let B [member of] [M.sub.m,r+1](S) be the matrix obtained by adding a zero column to the right of B and let [??] [member of] [M.sub.r+1,n](S) be the matrix obtained by adding a zero row to the bottom of C. Clearly [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now we compute the e-minors of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of order r + 1, using the Binet-Cauchy theorem; that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since the first summand is 0, so [I.sup.[epsilon].sub.r+1] is contained in the ideal generated by 1 [direct sum] [epsilon]. Hence [rank.sup.[epsilon].sub.det](A) [less than or equal to] r.

We can also prove a version of Sylvester's inequality for the ([epsilon], I)-rank.

Proposition 19. Let S be a commutative semiring and let [epsilon] be an element of S such that [[epsilon].sup.2] = 1 and that 1 [direct sum] [epsilon] is not a unit. Let I be an proper ideal of S which contains 1 [direct sum] [epsilon]. The inequality [rank.sup.[epsilon],I.sub.det] (AB) [less than or equal to] min([rank.sup.[epsilon],I.sub.det](A), [rank.sup.[epsilon],I.sub.det] (B)) holds for all A [member of] [M.sub.m,n](S) and B [member of] [M.sub.n,p](S).

Proof. Let r = min([rank.sup.[epsilon],I.sub.det] (A), [rank.sup.[epsilon],I.sub.det](B)). If r [greater than or equal to] min(m,n,p) we are done so suppose r < min(m, n, p) and let [alpha] and [beta] be both arbitrary subsets of {1,2, ..., n} of cardinality r+1. Then either [I.sup.[epsilon].sub.r+1](A) or [I.sup.[epsilon].sub.r+1] (B) is contained in I. It follows from the Binet-Cauchy theorem that [det.sub.[epsilon]](AB[[alpha], [beta]]) [member of] I.

It should be noted that the condition 1 [direct sum] [epsilon] [member of] I 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 S be a commutative semiring and let e be an element of S such that [[epsilon].sup.2] = 1 and that 1 [direct sum] [epsilon] is not a unit. Let I be an proper ideal of S which contains 1 [direct sum] [epsilon]. The inequality [rank.sup.[epsilon],I.sub.det] (A + B) [less than or equal to] [rank.sup.[epsilon],I.sub.det](A) + [rank.sup.[epsilon],I.sub.det](B) holds for all A, B [member of] [M.sub.m,n](S).

Proof. We begin by proving the inequality in the special case where m = n and [rank.sup.[epsilon],I.sub.det](A) + [rank.sup.[epsilon],I.sub.det](B) = n - 1. Hence A,B [member of] [M.sub.n](S). Let us suppose that r = [rank.sup.[epsilon],I.sub.det](A). This implies that n - r - 1 = [rank.sup.[epsilon],I.sub.det] (B). We can use Corollary 13 to show that [det.sub.[epsilon]] (A + B) [member of] I. Note that every term in the expansion of [det.sub.[epsilon]](A + B) is of a power of e times [det.sub.[epsilon]](A[[alpha] | [beta]])[cross product] [det.sub.[epsilon]] (B[[[alpha].sup.c] | [[beta].sup.c]]) where [alpha] and [beta] are subsets of {1,2, ..., n} satisfying [absolute value of [alpha]] = [absolute value of [beta]]. Let k = [absolute value of [alpha]] = [absolute value of [beta]]. If k [less than or equal to] r, then n - k [greater than or equal to] n - r > [rank.sup.[epsilon],I.sub.det](B) and since [absolute value of [[alpha].sup.c]] = [absolute value of [[beta].sup.c]] = we must have [det.sub.[epsilon]](B[[[alpha].sup.c] | [[beta].sup.c]]) [member of] I. Similarly, if k > r, then [det.sub.[epsilon]](A[[alpha] | [beta]]) [member of] I. Therefore every term in the expansion of [det.sub.[epsilon]](A + B) is in I and hence [rank.sup.[epsilon],I.sub.det](A + B) [less than or equal to] n - 1 = [rank.sup.[epsilon],I.sub.det] (A) + [rank.sup.[epsilon],I.sub.det] (B).

Now we prove the general case. Let r = [rank.sup.[epsilon],I.sub.det](A) + [rank.sup.[epsilon],I.sub.det](B). If r [greater than or equal to] min(m, n) then we are done so suppose r < min(m, n). Now let [alpha] and [beta] be subsets of {1,2, ..., m} and {1,2, ..., n}, respectively, both of cardinality r + 1. Then [rank.sup.[epsilon],I.sub.det] ((A + B)[[alpha] | [beta]]) < [rank.sup.[epsilon],I.sub.det] [[alpha] | [beta]]) + [rank.sup.[epsilon],I.sub.det] (B[[alpha] | [beta]]) < [rank.sup.[epsilon],I.sub.det](A) + [rank.sup.[epsilon],I.sub.det] (B) = r. Hence [det.sub.[epsilon]]((A+B)[[alpha] | [beta]]) [member of] I andsince (A + B)[[alpha] | [beta]] is an arbitrary r + 1 by r + 1 submatrix of A + B, we have [rank.sup.[epsilon],I.sub.det](A + B) [less than or equal to] r = [rank.sup.[epsilon],I.sub.det] (A) + [rank.sup.[epsilon],I.sub.det] (B).

3. Bijective Linear [epsilon]-Rank Preservers

In this section, we look at bijective linear operators which preserve ([epsilon], I)-rank of matrices over antinegative commutative semiring.

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

Proposition 22 (see [2]). For any commutative semiring, one has f(A) [less than or equal to] t(A) whenever A is an m byn matrix over S.

Let S be a semiring and A,B [member of] [M.sub.m,n](S). We write A [less than or equal to] B if there exists C [member of] [M.sub.m,n](S) such that A [direct sum] C = B. We note that the relation ([less than or equal to]) 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 T : [M.sub.m,n](S) [right arrow] [M.sub.m,n](S) preserves this preorder. Further, if S is an antinegative semiring then the term rank is a monotone function; that is, if A [less than or equal to] B then t(A) [less than or equal to] t(B).

Definition 23. Let S be a commutative semiring and A,B [member of] [M.sub.m,n](S). The Schur product of A and B, denoted as A [omicron] B, is an m by n matrix whose (i, j)th entry is [a.sub.ij] [cross product] [b.sub.ij].

Definition 24. Let S be a commutative semiring. A matrix A [member of] [M.sub.m,n](S) is called a submonomial matrix if every line (row or column) of A contains at most one nonzero entry. A matrix A [member of] [M.sub.n](S) is called a monomial matrix if every line (row or column) of A contains exactly one nonzero entry.

The concept of (P, Q, B) operator is a fundamental concept in the theory of linear preservers over semirings.

Definition 25 (see [13]). Let T be a linear operator from [M.sub.m,n](S) to itself. One says that T is a strong (P, Q, B) operator if there exist P [member of] [M.sub.m](S), Q [member of] [M.sub.n](S), and B [member of] [M.sub.m,n](S) such that P and Q are permutation matrices, and all of the entries of B are units and either T(X) = P(X [omicron] B)Q or m = n and T(X) = P([X.sup.T] [omicron] B)Q.

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 S be a commutative antinegative semiring. If rank : [M.sub.m,n](S) [right arrow] Z is a function which satisfies 0 [less than or equal to] rank(A) [less than or equal to] t(A) for all A [member of] [M.sub.m,n](S) with equality whenever A is a submonomial matrix, then any bijective linear operator which preserves this rank function must be a strong (P, Q, B) operator.

Since the ([epsilon], I)-rank satisfies the hypotheses of the above theorem, we now have the following corollary which classifies all bijective linear operators which preserve the ([epsilon], I)-rank.

Corollary 27. Let S be a commutative antinegative semiring and let e be an element of S such that [[epsilon].sup.2] = 1 and that 1 [direct sum] [epsilon] is not a unit. Let I be any proper ideal of S which contains 1 [direct sum] [epsilon]. Any bijective ([epsilon], I)-rank preserver on [M.sub.m,n](S) must be a strong (P, Q, B) operator.

4. Sign Pattern Matrices and the Sign Pattern Semiring

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

A matrix whose entries are from the set {+1,-1,0} is called a sign pattern matrix. If A = [[a.sub.ij]] is a real matrix, then the sign pattern of A is obtained from A, by replacing each entry by its signs [14,15]. The sign pattern of A is denoted by Sg(A) = [sg([a.sub.ij])], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

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 n by n sign pattern matrices by [Q.sub.n]. 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 {+1,-1,0, #}, where # corresponds to entries where the sign is unknown.

The set {+1,-1, 0, #} can be viewed as a semiring. If S = {+1,-1,0, #}, then (S, [direct sum], [cross product]) is a commutative semiring with identity, where the operations of addition and multiplication are defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Clearly all the properties of a semiring are satisfied where 0 is the additive identity and + 1 is multiplicative identity. Here + 1 and -1 are the units of S. More about the sign pattern semiring can be found in [17].

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

Definition 30 (see [14]). A sign pattern matrix A 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 e-rank. The sign pattern semiring has only two elements whose square is the identity, namely, 1 and -1. The ideal generated by 1 = 1 + 1 is the entire semiring but # = 1 + -1 generates the unique proper ideal {#, 0}. Therefore -1 is the only available choice for e and we have a unique e-rank. Hence [det.sub.[epsilon]](A) = [det.sub.+] (A) [direct sum] (-1 [cross product] det_(A)). It is easy to show that an n by n sign pattern matrix has [epsilon]-rank n if and only if it is an SNS matrix. Hence the [epsilon]-rank of a sign pattern matrix A is the largest integer k for which there exists a k by k SNS submatrix of A.

5. Sublocal Semirings

It was remarked earlier that [rank.sup.[epsilon],I.sub.det](A) [less than or equal to] [rank.sup.[epsilon],I.sub.det] (A). In other words amongst the family of ([epsilon], I)-ranks, choosing I to be the ideal generated by 1 [direct sum] [epsilon] gives us the largest possible rank function from this family. The minimal rank functions from this family arise from the choice of I to be a maximal ideal which contains 1 [direct sum] [epsilon]. In general, there maybe 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 I of a commutative semiring S is called a k-ideal if for any y [member of] S with x, x + y [member of] I one has y [member of] I.

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

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

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

Now we will define sublocal semirings using maximal ideals instead of k-maximal ideals. This is useful as some semirings do not have proper k-ideals. For example, the sign pattern semiring has only one proper ideal {0, #} and this is not a k-ideal.

Definition 34. Let S be a commutative semiring. One says that S is a sublocal semiring if S 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 I = {0, #}. We note that this maximal ideal is contained in the proper subsemiring P = {0, +1, #}; P however fails to be an ideal in the sign pattern semiring. The set of all natural numbers, N = {0,1,2, ...}, forms a sublocal semiring whose only one maximal ideal I = {N/{1}}. All chain semirings are sublocal semirings with S/{1} as a unique maximal ideal. A semifield is a commutative semiring in which all elements except 0 have a multiplicative inverse. (The Boolean and maxplus 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 a of a commutative semiring S is a unit of S if and only if a lies outside all maximal ideals of S.

Proof. Let a be a unit of S; then the ideal generated by a must be S itself and hence a lies outside all maximal ideals of S. If a is not a unit of S, then 1 [not member of] aS and hence there exits a maximal ideal M of S such that a [member of] aS [subset or equal to] M.

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

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

Since every k-ideal is an ideal, it follows that every local semiring is a sublocal semiring. The converse is false. Consider the nonnegative integers N with the usual addition and multiplication. The unique maximal ideal is N \ {1}; the maximal k-ideals are of the form pN for any prime p.

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 S be a sublocal semiring. Let [a.sub.1], [a.sub.2] [member of] S. If [[alpha].sub.1] [direct sum] [a.sub.2] is a unit of S, then either [a.sub.1] or [a.sub.2] is a unit of S.

6. Symmetrized Semirings

The ranks introduced in the previous sections all require an element e satisfying the condition that [[epsilon].sup.2] = 1 and 1 [direct sum] [epsilon] 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 [epsilon]. 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 (S, [direct sum], [cross product]) is a commutative semiring then [S.sup.2] = {(a, b) | a,b [member of] S} is also a commutative semiring with addition and multiplication defined as follows: for all a, b, c and d [member of] S,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

We can see that all the properties of a semiring are satisfied with (0, 0) being the additive identity of [S.sup.2]. Essentially this construction allows us to append an element [epsilon] = (0,1) with the property [[epsilon].sup.2] = 1 to the semiring S in a natural way giving us a way to apply the [epsilon]-determinant theory to semirings which do not have nontrivial self inversive elements. The ideal in [S.sup.2] generated by (1,1) = (1, 0) + [member of] is [DELTA] = {(x,x) : x [member of] S} which we will call the diagonal ideal. The [epsilon]-determinant in this case is the standard bideterminant and the e-rank is the standard determinantal rank defined as follows.

Definition 38. Let A be an m by n matrix over a commutative semiring S. The determinantal rank of A is the largest k for which there exists B,ak by k submatrix of A with [det.sub.+](B) [not equal to] [det.sub.-](B).

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 ([epsilon], I)-rank of matrices over general semirings.

Remark 39. Recall that [beta] denotes Boolean semiring which has two elements {0,1}. Then [[beta].sup.2] = {(0,0), (1, 0), (0,1), (1,1)} is also a semiring with the addition and the multiplication defined above for [S.sup.2]. Moreover it is isomorphic to the sign pattern semiring as (0,0) = 0, (1,0) = +1, (0,1) = -1, and (1,1) = #.

We complete our paper by showing that the symmetrized semiring [S.sup.2] inherits some important properties from S.

Theorem 40. Let S be a commutative semiring. If S is antinegative and has no zero divisors then [S.sup.2] is also antinegative and has no zero divisors.

Proof. Since S is antinegative, only 0 has an additive inverse. Let us suppose that ([a.sub.1], [b.sub.1]) [member of] [S.sup.2] has an additive inverse, so there exists ([a.sub.2], [b.sub.2]) [member of] [S.sup.2], such that ([a.sub.1],[b.sub.1]) [direct sum] ([a.sub.2], [b.sub.2]) = (0, 0). Consequently [a.sub.1] [direct sum] [a.sub.2] = 0 and [b.sub.1] [direct sum] [b.sub.2] = 0. Since S is antinegative so [a.sub.1] = [a.sub.2] = [b.sub.1] = [b.sub.2] = 0. Hence ([a.sub.1], [b.sub.1]) = (0, 0). Thus only the additive identity has an additive inverse in [S.sup.2] which means [S.sup.2] is an antinegative semiring. Now suppose that ([a.sub.1], [b.sub.1]) [member of] [S.sup.2] is a zero divisor, so there exists a nonzero ([a.sub.2], [b.sub.2]) [member of] [S.sup.2], such that ([a.sub.1], [b.sub.1]) [cross product] ([a.sub.2], [b.sub.2]) = (0,0). It follows that (([a.sub.1] [cross product] [a.sub.2]) [direct sum] ([b.sub.1] [cross product] [b.sub.2])) = 0 and (([a.sub.1] [cross product] [b.sub.2]) [direct sum] ([a.sub.2] [cross product][b.sub.1])) = 0. Since S is antinegative, [a.sub.1] [cross product][a.sub.2] = 0 and [b.sub.1] [cross product][b.sub.2] = 0. Also S has no zero divisors so either [a.sub.1] = 0 or [a.sub.2] = 0 and either [b.sub.1] = 0 or [b.sub.2] = 0; combining this with (([a.sub.1] [cross product] [b.sub.2]) [cross product] ([a.sub.2] [cross product] [b.sub.1])) = 0, we get either ([a.sub.1],[b.sub.1]) = (0, 0) or ([a.sub.2],[b.sub.2]) = (0,0). Hence [S.sup.2] has no zero divisors.

If S is a semiring, we let U(S) denote the set of units of S. There is a very close relation between the units of S and the units of [S.sup.2].

Lemma 41. If S is a commutative antinegative semiring with no zero divisors then U([S.sup.2]) = {(x, 0) : x [member of] U(S)}[union](0,x) : x [member of] U(S)}.

Proof. Let x be a unit in S. Then there exists a nonzero element a in S such that x [cross product] a = 1. Consider (x, 0) [cross product] (a, 0) = (x [cross product] a, 0) = (1, 0) and (0,x) [cross product] (0,a) = (x [cross product] a, 0) = (1, 0). Consequently (x, 0) and (0, x) are units of [S.sup.2]. Nowwe have to prove that these are the only units for [S.sup.2]. Suppose (x, y) [member of] [S.sup.2] is a unit in [S.sup.2]. Then there exists a nonzero element (a, b) [member of] [S.sup.2] such that (x, y) [cross product] (a, b) = (1,0). Thus ((a [cross product] x) [direct sum](b [cross product] y), (b [cross product] x) [direct sum] (a [cross product] y)) = (1, 0). Consequently (a [cross product] x) [direct sum] (b [cross product] y) = 1 and (b [cross product] x) [direct sum] (a [cross product] y) = 0. Since (b [cross product] x) [direct sum] (a [cross product] y) = 0 and S is an antinegative semiring, so b [cross product] x = 0 and a [cross product] y = 0. Also given that S has no zero divisors it follows that either b = 0 or x = 0 (note that both b and x cannot be zero because (a [cross product] x) [direct sum] (b [cross product] y) = 1) and either a = 0 or y = 0 (here also both a and y cannot be zero because (a[R] x) [R] (b [R] y) = 1). Since (x, y) and (a, b) are nonzero elements of S so the units of [S.sup.2], (x, y), have only two choices which are (x, 0) and (0, y). Putting (x,y) = (x, 0) in (a [cross product] x)[direct sum](b [cross product] y) = 1, we get a [cross product] x = 1, and this means that x is a unit of S. Putting (x, y) = (0, y) in (a [cross product] x) [direct sum] (b [cross product] y) = 1, we get b [cross product] y = 1, and this means that y is a unit of S. Thus all the units in [S.sup.2] are of the type (x, 0) and (0, x) where x is a unit in S.

We can now prove that if S is a sublocal antinegative semiring with no zero divisors then so is [S.sup.2].

Theorem 42. If S is a sublocal antinegative semiring with no zero divisors then [S.sup.2] is also a sublocal antinegative semiring with no zero divisors.

Proof. Suppose S is a sublocal antinegative semiring with no zero divisors; then by Theorem 40, [S.sup.2] is an antinegative semiring with no zero divisors. Hence we only need to prove that [S.sup.2] is sublocal; we show this by proving that the set of all nonunits in S forms an ideal of S . Let M = {(a, b) : where (a, b) is not a unit of [S.sup.2]} be the set of all nonunits of [S.sup.2]. Let ([a.sub.1], [b.sub.1]) and ([a.sub.2], [b.sub.2]) [member of] M such that ([a.sub.1], [b.sub.1]) [direct sum] ([a.sub.2],[b.sub.2]) = (x, 0), where x is a unit in S. Hence [a.sub.1] [direct sum] [a.sub.2] = x and [b.sub.1] [direct sum] [b.sub.2] = 0. Since S is antinegative so [b.sub.1] = 0 and [b.sub.2] = 0, and also [a.sub.1] [direct sum] [a.sub.2] = x, where x is a unit in S so (using Corollary 37) either [a.sub.1] is a unit or [a.sub.2] is a unit. Thus either ([a.sub.1], [b.sub.1]) = ([a.sub.1], 0), where [a.sub.1] is a unit in S, or ([a.sub.2],[b.sub.2]) = ([a.sub.2], 0), where [a.sub.2] is a unit in S. It follows that either ([a.sub.1], [b.sub.1]) is a unit or ([a.sub.2],[b.sub.2]) is a unit in [S.sup.2], which is a contradiction to the fact that both ([a.sub.1],[b.sub.1]) and ([a.sub.2],[b.sub.2]) [member of] M. Thus the sum of nonunits in [S.sup.2] is a nonunit. A similar argument works if ([a.sub.1],[b.sub.1])[direct sum]([a.sub.2], [b.sub.2]) = (0,x), where x is a unit in S. Now suppose that for (a, b) [member of] M and ([s.sub.1], [s.sub.2]) [member of] [S.sup.2] we have (a, b) [direct sum] ([s.sub.1], [s.sub.2]) = (x, 0), where x is a unit in S. Then (a [cross product] [s.sub.1]) [direct sum] (b [cross product] [s.sub.2]) = x and (a [cross product] [s.sub.2]) [direct sum] (b [cross product] [s.sub.1]) = 0. Since S is antinegative so a [cross product] [s.sub.2] = 0 and b [cross product] [s.sub.1] = 0, and also S has no zero divisors so either a = 0 or [s.sub.2] = 0 and either b = 0 or [s.sub.1] = 0. Clearly (a,b) or ([s.sub.1],[s.sub.2]) cannot be (0,0) since (a [cross product] [s.sub.1]) [direct sum] (b [cross product] [s.sub.2]) = x, so we have (a,b) = (0,b) or (a, 0). Further x is a unit in S and (a [cross product] [s.sub.1]) [direct sum] (b [cross product] [s.sub.2]) = x, so either a [cross product] [s.sub.1] is a unit in S or b [cross product] [s.sub.2] is a unit in S. Hence either a is a unit in S or b is a unit in S. We get (a, b) = (0, b) or (a, 0) is a unit in [S.sup.2], which is a contradiction to the fact that (a,b) [member of] M. Thus (a,b) [cross product] ([s.sub.1],[s.sub.2]) [member of] M for all (a,b) [member of] M and ([s.sub.1], [s.sup.2]) [member of] [S.sup.2]. Hence the set of all nonunits in [S.sup.2] forms an ideal of [S.sup.2]. Consequently [S.sup.2] is a sublocal semiring. A similar argument works if (a, b) [cross product] ([s.sub.1], [s.sub.2]) = (0, x), where x is a unit in S.

http://dx.doi.org/10.1155/2015/242515

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 (QEIIGSST). 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.

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Preeti Mohindru and Rajesh Pereira

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

Correspondence should be addressed to Rajesh Pereira; pereirar@uoguelph.ca

Received 28 November 2014; Accepted 9 January 2015

Academic Editor: Qing-Wen Wang
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