We use the

There are many equivalent ways of defining the rank of a matrix over a field. The rank of an

In this section, we review some background material on semirings. In Section

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 [

A semiring is a set

the additive identity

In other words, semirings are unital rings without the requirement that each element has additive inverse. If

The natural numbers form a semiring under the usual addition and multiplication. A much studied example of a semiring is

Let

If

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.

Let

As such we note that the determinant of a matrix

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.

Let

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

Let

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.

Let

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.

In [

Let

The term symmetry is from [

Let

Suppose

It follows easily from the previous proposition that if

We use this observation to slightly restate the definition of an

Let

We will get a distinct symmetry on

Tan has shown that the

Let

Let

In the case where

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

Let

The special case of this result where

We can use Tan’s generalization of the Laplace expansion to obtain a relationship between the

Let

Let

Let

It follows immediately from the Laplace expansion that

Let

For certain semirings, it may be possible that there is no choice of

The principal ideal generated by

Let

It is clear that any

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.

Let

The inequality

Let

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

Let

Let

It should be noted that the condition

We also have the following rank-sum inequality.

Let

We begin by proving the inequality in the special case where

Now we prove the general case. Let

In this section, we look at bijective linear operators which preserve

Let

For any commutative semiring, one has

Let

Let

Let

The concept of

Let

We also use a theorem from the same reference. We note though there is an error earlier in [

Let

Since the

Let

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

A matrix whose entries are from the set

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

The generalized sign pattern matrices are the matrices over the set

The set

Clearly all the properties of a semiring are satisfied where

Let

A sign pattern matrix

For matrices over the sign pattern semiring, we can give a more concrete interpretation of the

It was remarked earlier that

An ideal

We note that if we consider a commutative ring

A proper ideal

Let

Now we will define sublocal semirings using maximal ideals instead of

Let

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 [

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

An element

Let

One of the key results of [

A commutative semiring

Since every

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.

Let

The ranks introduced in the previous sections all require an element

If

Let

The determinantal rank has been much studied (see [

Recall that

We complete our paper by showing that the symmetrized semiring

Let

Since

If

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

Let

We can now prove that if

If

Suppose

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

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.