^{1}

^{2}

^{1}

^{2}

Let

A semiring is an algebra

A distributive lattice

In the following, we will introduce several kinds of distributive lattices which will often occur: general Boolean algebras (including binary Boolean algebras), chain semirings (including chains), and fuzzy semirings.

For a fixed positive integer

Let

Let

For a distributive lattice

The theory of matrices over distributive lattices has important applications in optimization theory, models of discrete event networks, and graph theory. There are a series of papers in the literature considering matrices over distributive lattices and similar topics (e.g., see [

It is well known that the decompositions of matrices over a distributive lattice play an important role in the studies of the lattice matrices. In this paper, we will firstly study the decompositions of matrices over a distributive lattice

For notations and terminologies that occurred but are not mentioned in this paper, readers are referred to [

Recall that an algebra

If a homomorphism

An element

Let

Lemma

By Lemma

Let

Let

In the following, we will study the decompositions of matrices over a distributive lattice

Assume that distributive lattice

Firstly, for any

Secondly, let

Finally, for any

In particular, in Theorem

Assume that distributive lattice

We only need to show that, for any

In fact, if we let

Now, analogous with the discussions of the above two theorems, the following theorem is also not hard to prove.

Assume that

Thus, we have the following.

Let

Let

Theorem

Let

Let

By Lemma

Now, for given

In particular, if we let finite distributive

Let

Also, consider finite distributive

Let

Summing up the above discussions, we have shown that a matrix over a finite distributive lattice

Let

Let

Let

By Lemma

Let

Given that

Note by Lemma

Let

Let

Similarly, let

Let

Theorem

As a direct application, we will firstly use the decompositions of matrices obtained in Section

Let

Now, by Theorem

Assume that

For any

Conversely, assume that

It is not hard to see that the indices and periods of the square matrices over a distributive lattice must exist. By Lemma

Let

In Theorem

Let

Theorem

In the following, consider

Let

Let

Let

Now, for given

Note by Lemma

On the other hand, by Theorem

Let

Similarly, if we take

Let

By applying Theorem

Let

Next, using the decompositions of matrices obtained in Section

Let

Let

For any

Conversely, assume that

For any

Let

By Theorem

Let

Let

Now, given

Note by Lemma

Let

Let

Similarly, let

Let

Theorem

In Theorem

Let

In particular, if we let

Let

On the other hand, by Theorem

Let

Let

Finally, we will use the decompositions of matrices obtained in Section

Let

Let

Assume that

Conversely, for

By Theorems

Let

By Theorem

Let

Let

For given

Note by Lemma

Let

Let

Similarly, let

Let

Theorem

In Theorem

Let

In particular, if we let

Let

On the other hand, by Theorem

Let

Let

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

The authors would like to thank the referees for their useful comments and suggestions contributed to this paper. This paper is supported by Grants of the National Natural Science Foundation of China (11261021 and 11226287); the Natural Science Foundation of Guangdong Province (S2012040007195); the Outstanding Young Innovative Talent Training Project in Guangdong Universities (2013LYM0086); the Key Discipline Foundation of Huizhou University.