Journal of Applied Mathematics

Volume 2014 (2014), Article ID 202075, 10 pages

http://dx.doi.org/10.1155/2014/202075

Research Article

On Decompositions of Matrices over Distributive Lattices

Yizhi Chen¹ and Xianzhong Zhao²

¹Department of Mathematics, Huizhou University, Huizhou, Guangdong 516007, China

²Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330027, China

Received 20 January 2014; Accepted 23 April 2014; Published 18 May 2014

Academic Editor: Jianming Zhan

Copyright © 2014 Yizhi Chen and Xianzhong Zhao. 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.

Let be a distributive lattice and ( , resp.) the semigroup (semiring, resp.) of (, resp.) matrices over . In this paper, we show that if there is a subdirect embedding from distributive lattice to the direct product of distributive lattices , then there will be a corresponding subdirect embedding from the matrix semigroup (semiring , resp.) to semigroup (semiring , resp.). Further, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of matrices over finite distributive lattices, chain semirings, fuzzy semirings, and so forth. Finally, as some applications, we present a method to calculate the indices and periods of the matrices over a distributive lattice and characterize the structures of idempotent and nilpotent matrices over it. We translate the characterizations of idempotent and nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which also generalize the corresponding structures of idempotent and nilpotent matrices over general Boolean algebras, chain semirings, fuzzy semirings, and so forth.

1. Introduction and Preliminaries

A semiring is an algebra with two binary operations + and such that both and are semigroups and such that the distributive laws

are satisfied. A partially ordered semiring means a semiring equipped with a compatible ordering ; that is, is a partial order on satisfying the following condition:

for any .

A distributive lattice is a lattice which satisfies either of the distributive laws and whose addition + and the multiplication on are as follows:

It is not hard to see that distributive lattice is a partially ordered semiring.

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 be the general Boolean algebra of subsets of a -element set and denote the singleton subsets of . Union is denoted by + and intersection by juxtaposition; 0 denotes the null set and 1 the set . Under these two operations, is a finite distributive lattice. In particular, if , is called the binary Boolean algebra.

Let be any set of two or more elements. If is totally ordered by (i.e., or for all distinct elements ), then define as max and as min for all . If has a universal lower bound and a universal upper bound, then becomes a semiring and is called a chain semiring.

Let be real numbers with . Define . Then, is a chain semiring with and . Furthermore, if we choose the real numbers 0 and 1 as and in the previous example, then is called fuzzy semiring.

For a distributive lattice , denote to be the set of all the matrices over . For any , define + in by

Then, clearly, is a semigroup. On the other hand, is denoted to be the set of all the matrices over . For any , the addition + in is defined as above, and the multiplication in is defined by

where . It is easy to verify that is also a semiring. We will call it the matrix semiring over and simply write it as .

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 [1–31]).

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 in Section 2. We show that if there is a subdirect embedding from distributive lattice to the direct product of distributive lattices , then there will be a corresponding subdirect embedding from the matrix semigroup (semiring , resp.) to semigroup (semiring , resp.). Further, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of matrices over finite distributive lattices, chain semirings, fuzzy semirings, and so forth, including the corresponding results in [6, 8, 30]. In Section 3, as some applications, we present a method to calculate the indices and periods of the matrices over a distributive lattice and study the structures of idempotent and nilpotent matrices over . We translate the characterizations of idempotent and nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which generalize and extend the corresponding structures of idempotent and nilpotent matrices over Boolean algebras, chain semirings, fuzzy semirings, and so forth.

For notations and terminologies that occurred but are not mentioned in this paper, readers are referred to [32–34].

2. Decompositions of Matrices over a Distributive Lattice

Recall that an algebra is said to be a subdirect product of an indexed family of algebras if it satisfies and for each , where is the projective mapping from to . And in this case, we also say that has a subdirect decomposition.

If a homomorphism from to is injective, then it is called an embedding. An embedding is called subdirect if is a subdirect product of the , and in this case we say that is isomorphic to a subdirect product of (see [32]).

An element of a lattice is called a join irreducible element of if implies or for . In a finite lattice , a nonzero element is join irreducible if and only if it has exactly one lower cover. Throughout this paper, the set of all join irreducible elements of will be denoted by .

Lemma 1 (see [6]). Let be a finite distributive lattice with elements and the set of all join irreducible elements of . Assume that is a subchain of such that and for ; define the mapping from to by

Then, is a subdirect embedding from lattice to lattice (i.e., lattice is isomorphic to a subdirect product of the .

Remark 2. Lemma 1 supplies us with a way to find subdirect decompositions for a finite distributive lattice into some of its subchains.

By Lemma 1, the following results are easy to obtain.

Lemma 3 (see [6]). Let be a finite distributive lattice with elements and the set of all join irreducible elements of . Take and define the mapping from to by

Then, is a subdirect embedding from lattice to lattice i.e., lattice is isomorphic to a subdirect product of the .

Lemma 4 (see [6]). Let be a chain with the usual ordering. Suppose that is any (but fixed) positive integer with and that is the least integer greater than or equal to . Take

Then, is isomorphic to a subdirect product of chains .

In the following, we will study the decompositions of matrices over a distributive lattice . Together with the subdirect decompositions of a finite distributive lattice obtained in the previous subsection, we will show that if there is a subdirect embedding from distributive lattice to the direct product of distributive lattices , then there will be a corresponding subdirect embedding from the matrix semigroup (semiring , resp.) to semigroup (semiring , resp.). And then we will prove that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains.

Theorem 5. Assume that distributive lattice is a subdirect product of distributive lattices . Define the mapping from to by

where and ( is the projective mapping from to ). Then, is a subdirect embedding from semigroup to semigroup .

Proof. Firstly, for any ,, we have

This shows that is injective.

Secondly, let . Then, we have

That is to say,

This implies .

Finally, for any , there exists such that

since is a surjection from to . That is to say, by taking , we have . This shows that is a subdirect embedding from semigroup to semigroup .

In particular, in Theorem 5, if we take , then we also have the following.

Theorem 6. Assume that distributive lattice is a subdirect product of distributive lattices . Define the mapping from to by

where and ( is the projective mapping from to ). Then, is a subdirect embedding from semiring to semiring .

Proof. We only need to show that, for any ,, .

In fact, if we let , then we have

That is to say,

This implies .

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

Theorem 7. Assume that is a subdirect embedding from distributive lattice to the direct product of distributive lattices . Define the mapping from (, resp.) to (, resp.) by

where and . Then, is a subdirect embedding from semigroup (semiring , resp.) to semigroup (semiring , resp.).

Thus, we have the following.

Theorem 8. Let be a finite distributive lattice with elements and the set of all join irreducible elements of . Assume that and are given as in Lemma 1. Define the mapping from , resp. to , resp. by

where and . Then, is a subdirect embedding from semigroup (semiring , resp.) to semigroup (semiring , resp.) and for any , resp.).

Proof. Let be a finite distributive lattice with elements and the set of all join irreducible elements of . Assume that and are given as in Lemma 1; that is, is a subchain of such that and for ; the mapping from to is defined by

We know by Lemma 1 that is a subdirect embedding from lattice to lattice . Thus, if we define the mapping from (, resp.) to (, resp.) by

where and , then it follows from Theorem 7 that is a subdirect embedding from semigroup (semiring , resp.) to semigroup (semiring , resp.). Also, it is easy to verify that, for any , resp.), . That is to say, .

Remark 9. Theorem 8 shows that a matrix over a finite distributive lattice can be decomposed into the sum of matrices over some special subchains of . That is to say, Theorem 7 generalizes Theorem 3.3 in [6]. Also, in Theorem 8, if we take and define , where

for any , then we can immediately obtain the following corollary.

Corollary 10. Let be a finite distributive lattice with elements and the set of all join irreducible elements of . Then, for any , the following holds:

Example 11. Let be a finite distributive lattice whose Hasse diagram is shown below

By Lemma 3, it is known that is isomorphic to the subdirect product of , , and . And then, by Theorem 8, is isomorphic to the subdirect product of , , and .

Now, for given , we have , where

In particular, if we let finite distributive to be general Boolean algebra in Corollary 10, we can get the following.

Corollary 12. Let be general Boolean algebra and the set of all join irreducible elements of . Then, for any , the following holds:

Also, consider finite distributive to be a chain with the usual ordering. Given the subchains of as in Lemma 4, define as in Theorem 8 and as in Theorem 3 in [30]. Then, is equal to . So we have the following.

Corollary 13. Let be a chain with the usual ordering. Then, for any , the following holds:

Summing up the above discussions, we have shown that a matrix over a finite distributive lattice can be decomposed into the sum of matrices over some of its special subchains. Furthermore, we can also obtain some general decompositions of a matrix over a distributive lattice .

Let be a distributive lattice. For any , is denoted to be the set of all the entries of ; that is, . Clearly, is a finite subset of . Also, denote to be the sublattice of generated by .

Lemma 14 (see [8]). Let be a distributive lattice. If is a finite subset of , then is a finite distributive sublattice of , where means a sublattice of generated by .

Theorem 15. Let be a distributive lattice. For any (, resp.), denote and . Assume that and are given as in Lemma 1. Define the mapping from (, resp.) to (, resp.) by

where and . Then, is a subdirect embedding from semigroup (semiring , resp.) to semigroup (semiring , resp.) and for any , resp.).

Proof. By Lemma 14, it is not hard to see that is a finite distributive lattice. Now, denote to be the set of all the join irreducible elements of . And then, analogous with the discussions of Theorem 8 or by Lemma 1 and Theorem 7, we can show the above theorem.

Example 16. Let be an infinite distributive lattice whose Hasse diagram is shown below.

Given that , clearly, .

Note by Lemma 1 that finite distributive lattice is isomorphic to a subdirect product of chains and , and then by Theorem 15 we have

Let be a chain semiring in Theorem 15. Then, for any , notice that for any ; we have the following corollary.

Corollary 17. Let be a chain semiring.   For  any  , denote and . Assume that and are given as in Lemma 1 Define the mapping from to by

where and . Then, is a subdirect embedding from semigroup to semigroup and for any .

Similarly, let be a fuzzy semiring in Theorem 15. Then, we will also have the following.

Corollary 18. Let be a fuzzy semiring. For any , denote and . Assume that and are given as in Lemma 1 Define the mapping from to by

where and . Then, is a subdirect embedding from semigroup to semigroup and for any .

Remark 19. Theorem 15 shows that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of matrices over finite distributive lattices, chain semirings, fuzzy semirings, and so forth, including the corresponding results in [6, 8, 30].

3. Some Applications

As a direct application, we will firstly use the decompositions of matrices obtained in Section 2 to give a way to calculate indices and periods of the square matrices over a distributive lattice.

Let be a distributive lattice and let be a semiring of matrices over . Recall that if there exist positive integers and satisfying for any (but fixed) , then the least such positive integers and are called the index and the period of , respectively, and denoted by and , respectively.

Now, by Theorem 7, it is not hard to obtain the following proposition.

Theorem 20. Assume that is a subdirect embedding from distributive lattice to the direct product of distributive lattices . Define the mapping from to by

where and . Then, for any and any positive integers and , one has

Proof. For any , assume that is a subdirect embedding from to with . If , then we have

that is,

and then

Thus, for .

Conversely, assume that , and, for , . Then, we have

that is,

and then

Notice that is the subdirect embedding from to ; we immediately get .

It is not hard to see that the indices and periods of the square matrices over a distributive lattice must exist. By Lemma 14, we can also get the following.

Theorem 21. Let be a distributive lattice. For any , denote , . Assume that and are given as in Lemma 1. Define the mapping from to by

where and . Then, the index of is equal to the maximum in the set of all indices of and the period of is equal to the least common multiple of all periods of .

In Theorem 21, take ; then, we have the following.

Corollary 22. Let be a distributive lattice. For any , denote , . Then, the index of is equal to the maximum in the set of all indices of and the period of is equal to the least common multiple of all periods of .

Remark 23. Theorem 21 and Corollary 22 show that the indices and periods of a square matrix over a distributive lattice can be calculated by studying some ways of subdirect decompositions of the corresponding matrix semirings.

In the following, consider to be a finite distributive lattice in Theorem 21. By Corollary 22, we immediately have the following.

Theorem 24. Let be a finite distributive lattice. For any , denote , . Assume that and are given as in Lemma 1. Define the mapping from to by

where and . Then, the index of is equal to the maximum in the set of all indices of and the period of is equal to the least common multiple of all periods of .

Corollary 25. Let be a finite distributive lattice. For any , denote , . Then, the index of is equal to the maximum in the set of all indices of and the period of is equal to the least common multiple of all periods of .

Example 26. Let be an infinite distributive lattice whose Hasse diagram is shown as in Example 16.

Now, for given then, .

Note by Lemma 1 that finite distributive lattice is isomorphic to a subdirect product of chains and , and then by Theorem 15 we have

It is easy to check that , and . Hence, .

On the other hand, by Theorem 20, we also have the following.

Theorem 27. Let be a finite distributive lattice with elements and the set of all join irreducible elements of . Assume that and are given as in Lemma 1. Define the mapping from to by

where and . Then, the index of is equal to the maximum in the set of all indices of and the period of is equal to the least common multiple of all periods of .

Similarly, if we take in Theorem 27, then we will get the following theorem.

Theorem 28. Let be a finite distributive lattice with elements and and the set of all join irreducible elements of . Then, the index of is equal to the maximum in the set of all indices of and the period of is equal to the least common multiple of all periods of .

By applying Theorem 27 to a finite chain , together with the discussions of Corollary 13, we can also immediately obtain the following theorem which had been proved by [30].

Theorem 29. Let be a finite chain with the usual ordering. For any , the index of is equal to the maximum in the set of all indices of   and the period of is equal to the least common multiple of all periods of.

Next, using the decompositions of matrices obtained in Section 2, we will give another application to calculate the idempotent matrices over a distributive lattice. We will translate the characterizations of idempotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which generalize and extend the corresponding structures of idempotent matrices over general Boolean algebras, chain semirings, fuzzy semirings, and so forth.

Let be a distributive lattice and a semiring of matrices over . Recall that a matrix in is called idempotent if .

Theorem 30. Let and be distributive lattices. Assume that is a subdirect embedding from semiring to semiring , and, for any , . Then, if and only if .

Proof. For any , assume that is a subdirect embedding from to with . If , then we have

that is,

and then

Thus, for .

Conversely, assume that , and for , . Then, we have

that is,

and then

Notice that is the subdirect embedding from to ; we immediately get .

For any , notice that is a finite distributive lattice; by Theorems 8 and 30, we immediately get the following.

Theorem 31. Let be a distributive lattice. For any , denote . Assume that are the subdistributive lattices of given as in Lemma 1 and is a subdirect embedding from semiring to semiring with . Then, if and only if   .

By Theorem 31, together with Theorem 8 or Corollary 10, we have the following.

Corollary 32. Let be a distributive lattice. For any , if and only if , where and .

Example 33. Let be an infinite distributive lattice whose Hasse diagram is shown as in Example 16.

Now, given , clearly,.

Note by Lemma 1 that finite distributive lattice is isomorphic to a subdirect product of chains and , and then, by Theorem 15, we have

By direct calculation (or by Theorem 2.11 in [14]), it is easy to check that

Thus, by Corollary 32,

Let be a chain semiring in Corollary 32. Then, for any , notice that for any ; we have the following corollary.

Corollary 34. Let be a matrix in . Then, is idempotent if and only if all are idempotent, where all are the nonzero join irreducible elements of and .

Similarly, let be a fuzzy semiring in Corollary 32. Then, we will also have the following.

Corollary 35. Let be a matrix in . Then, is idempotent if and only if all are idempotent, where all are the nonzero join irreducible elements of and .

Remark 36. Theorem 31 and Corollary 32 show that the distinctions of idempotent matrices over a general distributive lattice can be also translated into the distinctions of idempotent matrices over the binary Boolean cases, which generalize and extend the corresponding ones of idempotent matrices over general Boolean algebras and chain semirings (including the fuzzy semirings).

In Theorem 30, consider to be a finite distributive lattice. By Lemma 3 and Corollary 10, we immediately have the following.

Corollary 37. Let be a finite distributive lattice and be its all nonzero join irreducible elements. Then, for any , if and only if , where .

In particular, if we let be the general Boolean algebra in Corollary 37, then we will also obtain the following corollary which is just Theorem 4.1 in [24].

Corollary 38. Let be a matrix in with . Then, is idempotent if and only if all constituents of are idempotent in , where the constituent of is the matrix in whose entry is if and only if .

On the other hand, by Theorem 31, we will also have the following results.

Corollary 39. Let be a finite distributive lattice. For any , denote . Assume that are the subdistributive lattices of and is a subdirect embedding from semiring to semiring with . Then, if and only if .

Corollary 40. Let be a finite distributive lattice. For any , if and only if , where and .

Finally, we will use the decompositions of matrices obtained in Section 2 to calculate the nilpotent matrices over a distributive lattice. We will also translate the characterizations of nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which generalize and extend the corresponding structures of nilpotent matrices over general Boolean algebras, chain semirings, fuzzy semirings, and so forth.

Let be a distributive lattice and let be a semiring of matrices over . Recall that a matrix in is called nilpotent if for some , where is the zero matrix in .

Theorem 41. Let and be distributive lattices. Assume that is a subdirect embedding from semiring to semiring , and, for any , . Then, if and only if .

Proof. Assume that is a subdirect embedding from to and, for any , . If , then we have ; that is,

and then

Thus, we have for .

Conversely, for , if , then we have . Notice that is the subdirect embedding from to ; then, we immediately have .

By Theorems 8 and 41, we can get the following.

Theorem 42. Let be a distributive lattice. For any , denote . Assume that are the subdistributive lattices of given as in Lemma 1 and is a subdirect embedding from semiring to semiring with . Then, if and only if .

By Theorem 42, together with Theorem 8 or Corollary 10, we have the following.

Corollary 43. Let be a distributive lattice. For any , if and only if , where and .

Example 44. Let be an infinite distributive lattice whose Hasse diagram is shown as in Example 16.

For given , clearly,.

Note by Lemma 1 that finite distributive lattice is isomorphic to a subdirect product of chains and , and then, by Theorem 15, we have

By direct calculation, it is easy to check that

And then by Corollary 43,

Let be a chain semiring in Corollary 43. Then, for any , notice that for any ; we have the following corollary.

Corollary 45. Let be a matrix in . Then, is nilpotent if and only if all are nilpotent, where all are the nonzero join irreducible elements of and .

Similarly, let be a fuzzy semiring in Corollary 43. Then, we will also have the following.

Corollary 46. Let be a matrix in . Then, is nilpotent if and only if all are nilpotent, where all are the nonzero join irreducible elements of and .

Remark 47. Theorem 42 or Corollary 43 shows that the distinction of nilpotent matrices over a distributive lattice can be translated into the distinction of the binary Boolean cases, which generalize and extend the corresponding ones of nilpotent matrices over Boolean algebras and chain semirings (including the fuzzy semirings).

In Theorem 41, consider to be a finite distributive lattice. By Lemma 3 and Corollary 10, we immediately have the following.

Corollary 48. Let be a finite distributive lattice and let be its all nonzero join irreducible elements. Then, for any , if and only if , where .

In particular, if we let be the general Boolean algebra in Corollary 48, then we will obtain the following corollary.

Corollary 49. Let be a matrix in with . Then, is nilpotent if and only if all constituents of are nilpotent in , where the constituent of is the matrix in whose entry is if and only if .

On the other hand, by Theorem 42, we will also obtain the following corollaries.

Corollary 50. Let be a finite distributive lattice. For any , denote . Assume that are the subdistributive lattices of and is a subdirect embedding from semiring to semiring with . if and only if .

Corollary 51. Let be a finite distributive lattice. For any , if and only if , where and .

Conflict of Interests

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

Acknowledgments

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.

References

H. J. Bandelt and M. Petrich, “Subdirect products of rings and distributive lattices,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 25, no. 2, pp. 155–171, 1982.

R. B. Bapat, S. K. Jain, and L. E. Snyder, “Nonnegative idempotent matrices and the minus partial order,” Linear Algebra and its Applications, vol. 261, pp. 143–154, 1997.

L. B. Beasley and N. J. Pullman, “Linear operators strongly preserving idempotent matrices over semirings,” Linear Algebra and its Applications, vol. 160, pp. 217–229, 1992.

L. B. Beasley and A. E. Guterman, “The characterization of operators preserving primitivity for matrix -tuples,” Linear Algebra and its Applications, vol. 430, no. 7, pp. 1762–1777, 2009.

L. B. Beasley and S.-G. Lee, “Linear operators strongly preserving -potent matrices over semirings,” Linear Algebra and Its Applications, vol. 162–164, pp. 589–599, 1992.

Y.-Z. Chen, X.-Z. Zhao, and L. Yang, “On matrices over a finite distributive lattice,” Linear and Multilinear Algebra, vol. 60, no. 2, pp. 131–147, 2012.

R. P. Dilworth, “A decomposition theorem for partially ordered sets,” Annals of Mathematics. Second Series, vol. 51, pp. 161–166, 1950.

Z.-T. Fan, The Theory and Applications of Fuzzy Matrices, Science Publication, Beijing, China, 2006 (Chinese).

Z.-T. Fan and Q.-S. Cheng, “A survey on the powers of fuzzy matrices and FBAMs,” International Journal of Computational Cognition, vol. 2, pp. 1–25, 2004.

S. Ghosh, “A characterization of semirings which are subdirect products of a distributive lattice and a ring,” Semigroup Forum, vol. 59, no. 1, pp. 106–120, 1999.

S. Ghosh, “Matrices over semirings,” Information Sciences, vol. 90, no. 1–4, pp. 221–230, 1996.

Y. Give'on, “Lattice matrices,” Information and Computation, vol. 7, pp. 477–484, 1964.

H. Hashimoto, “Reduction of a nilpotent fuzzy matrix,” Information Sciences, vol. 27, no. 3, pp. 233–243, 1982.

K. T. Kang, S. Z. Song, and Y. O. Yang, “Structures of idempotent matrices over chain semirings,” Bulletin of the Korean Mathematical Society, vol. 44, no. 4, pp. 721–729, 2007.

K. H. Kim, Boolean Matrix Theory and Applications, vol. 70 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1982.

S. Kirkland and N. J. Pullman, “Boolean spectral theory,” Linear Algebra and its Applications, vol. 175, pp. 177–190, 1992.

S. Kirkland and N. J. Pullman, “Linear operators preserving invariants of nonbinary Boolean matrices,” Linear and Multilinear Algebra, vol. 33, no. 3-4, pp. 295–300, 1993.

J.-X. Li, “Some results on the nilpotent fuzzy matrices,” Fuzzy Systems and Mathematics, vol. 3, no. 1, pp. 52–55, 1989 (Chinese).

Y. Y. Lur, C. T. Pang, and S. M. Guu, “On simultaneously nilpotent fuzzy matrices,” Linear Algebra and its Applications, vol. 367, pp. 37–45, 2003.

Y. Y. Lur, C. T. Pang, and S. M. Guu, “On nilpotent fuzzy matrices,” Fuzzy Sets and Systems, vol. 145, no. 2, pp. 287–299, 2004.

M. Orel, “Nonbijective idempotents preservers over semirings,” Journal of the Korean Mathematical Society, vol. 47, no. 4, pp. 805–818, 2010.

O. A. Pshenitsyna, “Maps preserving invertibility of matrices over semirings,” Russian Mathematical Surveys, vol. 64, no. 1, pp. 162–164, 2009.

C. Reutenauer and H. Straubing, “Inversion of matrices over a commutative semiring,” Journal of Algebra, vol. 88, no. 2, pp. 350–360, 1984.

S. Z. Song, K. T. Kang, and L. B. Beasley, “Idempotent matrix preservers over Boolean algebras,” Journal of the Korean Mathematical Society, vol. 44, no. 1, pp. 169–178, 2007.

S. Z. Song, K. T. Kang, L. B. Beasley, and N. S. Sze, “Regular matrices and their strong preservers over semirings,” Linear Algebra and its Applications, vol. 429, no. 1, pp. 209–223, 2008.

Y.-J. Tan, “On the powers of matrices over a distributive lattice,” Linear Algebra and its Applications, vol. 336, pp. 1–14, 2001.

Y.-J. Tan, “On compositions of lattice matrices,” Fuzzy Sets and Systems, vol. 129, no. 1, pp. 19–28, 2002.

Y.-J. Tan, “On nilpotent matrices over distributive lattices,” Fuzzy Sets and Systems, vol. 151, no. 2, pp. 421–433, 2005.

Y.-J. Tan, “On the transitive matrices over distributive lattices,” Linear Algebra and its Applications, vol. 400, pp. 169–191, 2005.

X.-Z. Zhao, Y. B. Jun, and F. Ren, “The semiring of matrices over a finite chain,” Information Sciences, vol. 178, no. 17, pp. 3443–3450, 2008.

K.-L. Zhang, “On the nilpotent matrices over -lattice,” Fuzzy Sets and Systems, vol. 117, no. 3, pp. 403–406, 2001.

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition, 2000.

J. S. Golan, Semirings and Their Applications, Kluwer Academic, Dordrecht, The Netherlands, 1999.

J. M. Howie, Fundamentals of Semigroup Theory, vol. 12 of London Mathematical Society Monographs. New Series, The Clarendon Press, Oxford, UK, 1995.