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

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.


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][2][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 0, 1 in with 0 ̸ = 1, such that (1) ( , ⊕, 0) is a commutative monoid, (2) ( , ⊗, 1) is a monoid, (3) ⊗ is both left and right distributive over ⊕, In other words, semirings are unital rings without the requirement that each element has additive inverse. If ( , ⊗, 1) is a commutative monoid then is called a commutative 2 International Scholarly Research Notices 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 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 (R max , ⊕, ⊗), where R max = R ⋃{−∞} with ⊕ = max{ , } and ⊗ = + . In this case 0 = −∞ and 1 = 0 [8]. A totally ordered set with greatest element 1 and least element 0 forms a semiring with ⊕ = max{ , } and ⊗ = min{ , }. This is called chain semiring. The chain semiring with two elements {0, 1} is called the Boolean semiring and it is denoted by . 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 sgn( ) = +1 if is even permutation and sgn( ) = −1 if is odd permutation. Here sgn( ) 1 (1) 2 (2) ⋅ ⋅ ⋅ ( ) 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.
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.

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].
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 (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. 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.
We will get a distinct symmetry on and hence a distinct det from every choice of ∈ which satisfies 2 = 1. One candidate for that exists in every semiring is the multiplicative identity 1. If = 1, we get det ( ) = per( ).
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 denote the by submatrix of whose ( , )th entry is , . Theorem 11 (see [5,Theorem 3.3] (the generalized Laplace expansion)). Let be a commutative semiring and let ∈ satisfy 2 = 1. If ∈ ( ) and ⊆ {1, 2, . . . , }, then In the case where is a commutative ring and = −1, 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
The special case of this result where is a Boolean algebra (and by necessity = 1 which means det 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 ∈ N, let be a commutative semiring, and let ∈ satisfy 2 = 1. The identity 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 thedeterminant, det ( + ) = ⨁ ⊆{1,2,..., } det ( ). Now we use the Laplace expansion on det ( ), for any fixed , to get Hence Definition 14. Let be a commutative semiring and let be an by matrix over . If 1 ≤ ≤ min( , ), let ( ) be the ideal in generated by the set of all the by -minors of . One defines 0 ( ) = and ( ) = {0} when > min( , ).

It follows immediately from the Laplace expansion that
Definition 15. Let be a commutative semiring and let be an element of such that 2 = 1 and that 1 ⊕ is not a unit. Let be any proper ideal of which contains 1 ⊕ . The ( , )-rank of an by matrix (denoted by rank , det ( )) 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 1 ⊕ is not a unit. An example of this is the semiring of nonnegative real numbers R + with the usual addition and multiplication. In this case the only ∈ R + satisfying 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 ( , )-rank. We will examine how to handle cases like this in the last section of the paper.
The principal ideal generated by 1 ⊕ is a natural choice for our ideal . International Scholarly Research Notices Definition 16. Let be a commutative semiring and let be an element of such that 2 = 1 and that 1 ⊕ is not a unit. Let 1⊕ be the principal ideal generated by 1 ⊕ . The -rank of an by matrix (denoted by rank det ( )) is the largest nonnegative integer such that ( ) is not contained in 1⊕ .
It is clear that any containing 1 ⊕ contains 1⊕ and therefore rank , det ( ) ≤ rank det ( ). 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 ( ).
Proof. Let = ( ), then there exist matrices ∈ , ( ) and ∈ , ( ) such that = . Let̂∈ , +1 ( ) be the matrix obtained by adding a zero column to the right of and let̂∈ +1, ( ) be the matrix obtained by adding a zero row to the bottom of . Clearly =̂̂. Now we compute the -minors of (=̂̂) of order + 1, using the Binet-Cauchy theorem; that is, det 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 2 = 1 and that 1 ⊕ is not a unit. Let be an proper ideal of which contains 1 ⊕ . The inequality rank , det ( ) ≤ min(rank , det ( ), rank , det ( )) holds for all ∈ , ( ) and ∈ , ( ).
It should be noted that the condition 1 ⊕ ∈ 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 2 = 1 and that 1 ⊕ is not a unit. Let be an proper ideal of which contains 1 ⊕ . The inequality rank , det ( + ) ≤ rank , det ( ) + rank , det ( ) holds for all , ∈ , ( ).
Proof. We begin by proving the inequality in the special case where = and rank , det ( ) + rank , det ( ) = − 1. Hence , ∈ ( ). Let us suppose that = rank , det ( ). This implies that − − 1 = rank , det ( ). We can use Corollary 13 to show that det ( + ) ∈ . Note that every term in the expansion of det

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 ( ).
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.
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.
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 2 = 1 and that 1 ⊕ is not a unit. Let be any proper ideal of which contains 1 ⊕ . Any bijective ( , )-rank preserver on , ( ) must be a strong ( , , ) operator.

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 {+1, −1, 0} 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 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.
The set {+1, −1, 0, #} can be viewed as a semiring. If = {+1, −1, 0, #}, 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 0 is the additive identity and +1 is multiplicative identity. Here +1 and −1 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, 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 and we have a unique -rank. Hence det ( ) = det + ( ) ⊕ (−1 ⊗ det − ( )). It is easy to show that an by sign pattern matrix has -rank if and only if it is an SNS matrix. Hence therank of a sign pattern matrix is the largest integer for which there exists a by SNS submatrix of .

Sublocal Semirings
It was remarked earlier that rank , det ( ) ≤ rank det ( ). In other words amongst the family of ( , )-ranks, choosing to be the ideal generated by 1 ⊕ 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 1 ⊕ . 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.

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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 semiringideals 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 {0, #} 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 = {0, #}. We note that this maximal ideal is contained in the proper subsemiring = {0, +1, #}; 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 = {N/{1}}. All chain semirings are sublocal semirings with /{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 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 1 ∉ 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 N with the usual addition and multiplication. The unique maximal ideal is N \ {1}; the maximal -ideals are of the form N 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 1 , 2 ∈ . If 1 ⊕ 2 is a unit of , then either 1 or 2 is a unit of .

Symmetrized Semirings
The ranks introduced in the previous sections all require an element satisfying the condition that 2 = 1 and 1 ⊕ 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 2 = {( , ) | , ∈ } 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 (0, 0) being the additive identity of 2 . Essentially this construction allows us to append an element = (0, 1) with the property 2 = 1 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 2 generated by (1, 1) = (1, 0) + is Δ = {( , ) : ∈ S} 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 det + ( ) ̸ = det − ( ).
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.
We complete our paper by showing that the symmetrized semiring 2 inherits some important properties from .
Theorem 40. Let be a commutative semiring. If is antinegative and has no zero divisors then 2 is also antinegative and 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 2 . is an antinegative semiring, so ⊗ = 0 and ⊗ = 0. Also given that has no zero divisors it follows that either = 0 or = 0 (note that both and cannot be zero because ( ⊗ ) ⊕ ( ⊗ ) = 1) and either = 0 or = 0 (here also both and cannot be zero because ( ⊗ ) ⊕ ( ⊗ ) = 1). Since ( , ) and ( , ) are nonzero elements of 2 so the units of 2 , ( , ), have only two choices which are ( , 0) and (0, ). Putting ( , ) = ( , 0) in ( ⊗ )⊕( ⊗ ) = 1, we get ⊗ = 1, and this means that is a unit of . Putting ( , ) = (0, ) in ( ⊗ ) ⊕ ( ⊗ ) = 1, we get ⊗ = 1, and this means that is a unit of . Thus all the units in 2 are of the type ( , 0) and (0, ) where is a unit in .
We can now prove that if is a sublocal antinegative semiring with no zero divisors then so is 2 .
Theorem 42. If is a sublocal antinegative semiring with no zero divisors then 2 is also a sublocal antinegative semiring with no zero divisors.