# On Decompositions of Matrices over Distributive Lattices

Authors: Yizhi Chen; Xianzhong Zhao
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

LetL be a distributive lattice and Mn,q (L)(Mn(L), resp.) the semigroup (semiring, resp.) of n×q (n×n, resp.) matrices over L. In this paper, we show that if there is a subdirect embedding from distributive lattice L to the direct product ∏i=1m‍Li of distributive lattices L1,L2,…,Lm, then there will be a corresponding subdirect embedding from the matrix semigroup Mn,q(L) (semiring Mn(L), resp.) to semigroup ∏i=1m‍Mn,q(Li) (semiring ∏i=1m‍Mn(Li), 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.


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## Body

## 1. Introduction and Preliminaries

A semiring is an algebra(R,+,·) with two binary operations + and · such that both (R,+) and (R,·) are semigroups and such that the distributive laws
(1)x(y+z)≈xy+xz,(x+y)z≈xz+yz
are satisfied. A partially ordered semiring means a semiring R equipped with a compatible ordering ≤; that is, ≤ is a partial order on R satisfying the following condition:
(2)a≤b,c≤d⟹a+c≤b+d,ac≤bd
for any a,b,c,d∈R.A distributive latticeL is a lattice which satisfies either of the distributive laws and whose addition + and the multiplication · on L are as follows:
(3)a+b=a∨b,ab=a∧b.
It is not hard to see that distributive lattice L 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 integerk, let Bk be the general Boolean algebra of subsets of a k-element set Sk andσ1,σ2,…,σk denote the singleton subsets of Sk. Union is denoted by + and intersection by juxtaposition; 0 denotes the null set and 1 the set Sk. Under these two operations, Bk is a finite distributive lattice. In particular, if k=1, B1 is called the binary Boolean algebra.LetK be any set of two or more elements. If K is totally ordered by < (i.e., x<y or y<x for all distinct elements x,y∈K), then define x+y as max{x,y} and xy as min{x,y} for all x,y∈K. If K has a universal lower bound and a universal upper bound, then K becomes a semiring and is called a chain semiring.Letα,ω be real numbers with α<ω. Define S={β∈R:α≤β≤ω}. Then, S is a chain semiring with α=0 and ω=1. Furthermore, if we choose the real numbers 0 and 1 as α and ω in the previous example, then F={β∈R:0≤β≤1} is called fuzzy semiring.For a distributive latticeL, denote Mn,p(L) to be the set of all the n×p matrices over L. For any A=(aij),B=(bij)∈Mn,p(L), define + in Mn,p(L) by
(4)A+B=(aij+bij).
Then, clearly, (Mn,p(L),+) is a semigroup. On the other hand, Mn(L) is denoted to be the set of all the n×n matrices over L. For any A=(aij),B=(bij)∈Mn(L), the addition + in Mn(L) is defined as above, and the multiplication · in Mn(L) is defined by
(5)AB=(cij),
where cij=∑k=1n‍aikbkj. It is easy to verify that (Mn(L),+,·) is also a semiring. We will call it the n×n matrix semiring over L and simply write it as Mn(L).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 latticeL in Section 2. We show that if there is a subdirect embedding from distributive lattice L to the direct product ∏i=1m‍Li of distributive lattices L1,L2,…,Lm, then there will be a corresponding subdirect embedding from the matrix semigroup Mn,q(L) (semiring Mn(L), resp.) to semigroup ∏i=1m‍Mn,q(Li) (semiring ∏i=1m‍Mn(Li), 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 L. 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 algebraA is said to be a subdirect product of an indexed family Ai(i∈I) of algebras if it satisfies A≤∏i∈I‍Ai and Aπi=Ai for each i∈I, where πi is the projective mapping from A to Ai. And in this case, we also say that A has a subdirect decomposition.If a homomorphismφ from A to ∏i∈I‍Ai is injective, then it is called an embedding. An embedding φ:A→∏i∈I‍Ai is called subdirect if Aφ is a subdirect product of the Ai, and in this case we say that A is isomorphic to a subdirect product of Ai (see [32]).An elementa of a lattice L is called a join irreducible element of L if a=x+y implies x=a or y=a for x,y∈L. In a finite lattice L, 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 L will be denoted by J.Lemma 1 (see [6]).
LetL be a finite distributive lattice with l≥2 elements and J={0,p1,p2,…,ph} the set of all join irreducible elements of L. Assume that Ci(i=1,2,…,m) is a subchain 0=pi0<pi1<⋯<pisi of J such that J=⋃i=1m‍Ci and Ci⋂‍Cj={0} for i≠j; define the mapping ψ from L to ∏i=1m‍Ci by
(6)(i)(a)ψ={pisia≥pisi,pika≥pikbuta≱pi(k+1),iiiiiiiiiiiii(∀a∈L,1≤i≤m,1≤k<si).
Then, ψ is a subdirect embedding from lattice L to lattice ∏i=1m‍Ci (i.e., lattice L is isomorphic to a subdirect product of the Ci).Remark 2.
Lemma1 supplies us with a way to find subdirect decompositions for a finite distributive lattice L into some of its subchains.By Lemma1, the following results are easy to obtain.Lemma 3 (see [6]).
LetL be a finite distributive lattice with l≥2 elements and J={0,p1,p2,…,ph} the set of all join irreducible elements of L. Take Ci={0,pi}(i=1,2,…,h) and define the mapping ψ from L to ∏i=1h‍Ci by
(7)(i)(a)ψ={pia≥pi,0otherwise.(∀a∈L,1≤i≤h)
Then, ψ is a subdirect embedding from lattice L to lattice ∏i=1h‍Ci(i.e., lattice L is isomorphic to a subdirect product of the Ci).Lemma 4 (see [6]).
LetL={0,1,…,l}(l>1) be a chain with the usual ordering. Suppose that s is any (but fixed) positive integer with 1<s<l and that m=⌈l/s⌉ is the least integer greater than or equal to l/s. Take
(8)Ci={0,s(i-1)+1,s(i-1)+2,…,si}iiiiiiiiiiiiiiiiiiiiiiiiiii(i=1,2,…,m-1),Cm={0,s(m-1)+1,s(m-1)+2,…,l}.
Then, L is isomorphic to a subdirect product of chains Ci(i=1,2,…,m).In the following, we will study the decompositions of matrices over a distributive latticeL. 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 L to the direct product ∏i=1m‍Li of distributive lattices L1,L2,…,Lm, then there will be a corresponding subdirect embedding from the matrix semigroup Mn,q(L) (semiring Mn(L), resp.) to semigroup ∏i=1m‍Mn,q(Li) (semiring ∏i=1m‍Mn(Li), 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 latticeL is a subdirect product of distributive lattices L1,L2,…,Lm. Define the mapping φ from Mn,q(L) to ∏i=1m‍Mn,q(Li) by
(9)(∀A=(aij)∈Mn,q(L))Aφ=(A1,A2,…,Am),where(∀1≤k≤m)Ak=(aij(k)) and aij(k)=(aij)πk(πk is the projective mapping from L to Lk).Then,φis a subdirect embedding from semigroupMn,q(L) to semigroup ∏i=1m‍Mn,q(Li).Proof.
Firstly, for anyA=(aij),B=(bij)∈Mn,q(L), we have
(10)Aφ=Bφ⟺(∀1≤k≤m)Ak=Bk⟺(∀1≤k≤m,∀1≤i,j≤n)aij(k)=bij(k)⟺(∀1≤k≤m,∀1≤i,j≤n)⟺(aij)πk=(bij)πk⟺(∀1≤i,j≤n)aij=bij⟺A=B.
This shows that φ is injective.
Secondly, letC=(cij)=A+B. Then, we have
(11)cij(k)=(aij+bij)πk=(aij)πk+(bij)πk=aij(k)+bij(k)iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii(∀1≤k≤m,1≤i,j≤n).
That is to say,
(12)Ck=Ak+Bk(∀1≤k≤m).
This implies (A+B)φ=Aφ+Bφ.
Finally, for anyB=(bij)∈Mn,q(Lk)(1≤k≤m), there exists aij∈L such that
(13)(aij)πk=bij,
since πk is a surjection from L to Lk. That is to say, by taking A=(aij), we have (Aφ)πk=B. This shows that φ is a subdirect embedding from semigroup Mn,q(L) to semigroup ∏i=1m‍Mn,q(Li).In particular, in Theorem5, if we take n=q, then we also have the following.Theorem 6.
Assume that distributive latticeL is a subdirect product of distributive lattices L1,L2,…,Lm. Define the mapping φ from Mn(L) to ∏i=1m‍Mn(Li) by
(14)(∀A=(aij)∈Mn(L))Aφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(aij)πk (πk is the projective mapping from L to Lk). Then, φ is a subdirect embedding from semiring Mn(L) to semiring ∏i=1m‍Mn(Li).Proof.
We only need to show that, for anyA=(aij),B=(bij)∈Mn(L), (AB)φ=(Aφ)(Bφ).
In fact, if we letD=(dij)=AB, then we have
(15)dij(k)=(∑l=1n‍ailblj)πk=∑l=1n‍(ail)πk(blj)πk=∑l=1n‍ail(k)blj(k)iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii(∀1≤k≤m,1≤i,j≤n).
That is to say,
(16)Dk=AkBk(∀1≤k≤m).
This implies (AB)φ=(Aφ)(Bφ).Now, analogous with the discussions of the above two theorems, the following theorem is also not hard to prove.Theorem 7.
Assume thatψ is a subdirect embedding from distributive lattice L to the direct product ∏i=1m‍Li of distributive lattices L1,L2,…,Lm. Define the mapping φ from Mn,q(L) (Mn(L), resp.) to ∏i=1m‍Mn,q(Li) (∏i=1m‍Mn(Li), resp.) by
(17)(∀A=(aij)∈Mn,q(L))(Mn(L),resp.)iiiiiiiiiiiiiiiiiiiiiiiAφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ. Then, φ is a subdirect embedding from semigroup Mn,q(L) (semiring Mn(L), resp.) to semigroup ∏i=1m‍Mn,q(Li) (semiring ∏i=1m‍Mn(Li), resp.).Thus, we have the following.Theorem 8.
LetL be a finite distributive lattice with l≥2 elements and J={0,p1,p2,…,ph} the set of all join irreducible elements of L. Assume that Ci(i=1,2,…,m) and ψ are given as in Lemma 1. Define the mapping φ from Mn,q(L)(Mn(L), resp.) to ∏i=1m‍Mn,q(Ci)(∏i=1m‍Mn(Ci), resp.) by
(18)(∀A=(aij)∈Mn,q(L))(Mn(L),resp.)iiiiiiiiiiiiiiiiiiiiiiiiAφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ. Then, φ is a subdirect embedding from semigroup Mn,q(L) (semiring Mn(L), resp.) to semigroup ∏i=1m‍Mn,q(Ci) (semiring ∏i=1m‍Mn(Ci), resp.) and A=∑k=1m‍Ak for any A∈Mn,q(L)(Mn(L), resp.).Proof.
LetL be a finite distributive lattice with l≥2 elements and J={0,p1,p2,…,ph} the set of all join irreducible elements of L. Assume that Ci(i=1,2,…,m) and ψ are given as in Lemma 1; that is, Ci(i=1,2,…,m) is a subchain 0=pi0<pi1<⋯<pisi of J such that J=⋃i=1m‍Ci and Ci⋂‍Cj={0} for i≠j; the mapping ψ from L to ∏i=1m‍Ci is defined by
(19)(i)(a)ψ={pisia≥pisi,pika≥pikbuta≱pi(k+1),iiiiiiiiiii(∀a∈L,1≤i≤m,1≤k<si).
We know by Lemma 1 that ψ is a subdirect embedding from lattice L to lattice ∏i=1m‍Ci. Thus, if we define the mapping φ from Mn,q(L) (Mn(L), resp.) to ∏i=1m‍Mn,q(Ci)(∏i=1m‍Mn(Ci), resp.) by
(20)(∀A=(aij)∈Mn,q(L))(Mn(L),resp.)iiiiiiiiiiiiiiiiiiiiiiiiiAφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ, then it follows from Theorem 7 that φ is a subdirect embedding from semigroup Mn,q(L) (semiring Mn(L), resp.) to semigroup ∏i=1m‍Mn,q(Ci) (semiring ∏i=1m‍Mn(Ci), resp.). Also, it is easy to verify that, for any A=(aij)∈Mn,q(L)(Mn(L), resp.), aij=∑k=1m‍aij(k). That is to say, A=∑k=1m‍Ak.Remark 9.
Theorem8 shows that a matrix over a finite distributive lattice L can be decomposed into the sum of matrices over some special subchains of L. That is to say, Theorem 7 generalizes Theorem 3.3 in [6]. Also, in Theorem 8, if we take Ci={0,pi}(1≤i≤h) and define Api=(ckl), where
(21)ckl={1,ifakl≥pi,0,else,
for any 1≤k≤n,1≤l≤h, then we can immediately obtain the following corollary.Corollary 10.
LetL be a finite distributive lattice with l≥2 elements and J={0,p1,p2,…,ph} the set of all join irreducible elements of L. Then, for any A=∈Mn,q(L), the following holds:
(22)A=∑i=1h‍piApi.Example 11.
LetL={0,a,b,c,1} be a finite distributive lattice whose Hasse diagram is shown below
By Lemma3, it is known that L is isomorphic to the subdirect product of C1={0,a}, C2={0,b}, and C3={0,1}. And then, by Theorem 8, Mn,p(L) is isomorphic to the subdirect product of Mn,p(C1), Mn,p(C2), and Mn,p(C3).
Now, for givenA=(ab1cbbaa1abc)∈M3,4(L), we have A=A1+A2+A3, where
(23)A1=(a00a00aa0a0a)∈M3,4(C1),A2=(0b0bbb0000bb)∈M3,4(C2),A3=(001000001000)∈M3,4(C3).In particular, if we let finite distributiveL to be general Boolean algebra Bk in Corollary 10, we can get the following.Corollary 12.
LetBk be general Boolean algebra and J={0,p1,p2,…,ph} the set of all join irreducible elements of Bk. Then, for any A∈Mn,q(Bk), the following holds:
(24)A=∑i=1h‍piApi.Also, consider finite distributiveL to be a chain {0,1,…,l}(l>1) with the usual ordering. Given the subchains Ci(i=1,2,…,m) of L as in Lemma 4, define Ak as in Theorem 8 and ψ~sl(k)φ-ls(k)(A) as in Theorem 3 in [30]. Then, Ak is equal to ψ~sl(k)φ-ls(k)(A). So we have the following.Corollary 13.
LetL={0,1,…,l}(l>1) be a chain with the usual ordering. Then, for any A∈Mn(L), the following holds:
(25)A=∑s=1m‍ψ~sl(k)φ-ls(k)(A).Summing up the above discussions, we have shown that a matrix over a finite distributive latticeL 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 L.LetL be a distributive lattice. For any A=(aij)∈Mn,p(L), ΦA is denoted to be the set of all the entries of A; that is, ΦA={aij∣1≤i≤n,1≤j≤p}. Clearly, ΦA={aij∣1≤i≤n,1≤j≤p} is a finite subset of L. Also, denote 〈ΦA〉 to be the sublattice of L generated by ΦA.Lemma 14 (see [8]).
LetL be a distributive lattice. If S is a finite subset of L, then 〈S〉 is a finite distributive sublattice of L, where 〈S〉 means a sublattice of L generated by S.Theorem 15.
LetL be a distributive lattice. For any A∈Mn,q(L)(Mn(L), resp.), denote 〈ΦA〉=L- and J〈ΦA〉={0,p1,p2,…,ph}. Assume that Ci(i=1,2,…,m) and ψ are given as in Lemma 1. Define the mapping φ from Mn,q(L-)(Mn(L-), resp.) to ∏i=1m‍Mn,q(Ci) (∏i=1m‍Mn(Ci), resp.) by
(26)(∀A=(aij)∈Mn,q(L-))(Mn(L-),resp.)iiiiiiiiiiiiiiiiiiiiiiiiAφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ. Then, φ is a subdirect embedding from semigroup Mn,q(L-) (semiring Mn(L-), resp.) to semigroup ∏i=1m‍Mn,q(Ci) (semiring ∏i=1m‍Mn(Ci), resp.) and A=∑k=1m‍Ak for any A∈Mn,q(L)(Mn(L), resp.).Proof.
By Lemma14, it is not hard to see that 〈ΦA〉 is a finite distributive lattice. Now, denote J〈ΦA〉 to be the set of all the join irreducible elements of 〈ΦA〉. And then, analogous with the discussions of Theorem 8 or by Lemma 1 and Theorem 7, we can show the above theorem.Example 16.
LetL={0,a,b,c,d,…} be an infinite distributive lattice whose Hasse diagram is shown below.
Given thatA=(caaabb0ca0aa)∈M3,4(L), clearly, L-=〈ΦA〉={0,a,b,c},J〈ΦA〉={0,a,b}.
Note by Lemma1 that finite distributive lattice L-={0,a,b,c} is isomorphic to a subdirect product of chains {0,a} and {0,b}, and then by Theorem 15 we have
(27)A=(caaabb0ca0aa)=(aaaa000aa0aa)+(b000bb0b0000).LetL=K be a chain semiring in Theorem 15. Then, for any A∈Mn(K), notice that 〈ΦA〉=ΦA for any A∈Mn(K); we have the following corollary.Corollary 17.
LetK be a chain semiring.   For  any  A∈Mn,q(K), denote ΦA=L- and JΦA={0,p1,p2,…,ph}. Assume that Ci(i=1,2,…,m) and ψ are given as in Lemma 1 Define the mapping φ from Mn,q to ∏i=1m‍Mn,q(Ci) by
(28)(∀A=(aij)∈Mn,q(L-))Aφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ. Then, φ is a subdirect embedding from semigroup Mn,q(L-) to semigroup ∏i=1m‍Mn,q(Ci) and A=∑k=1m‍Ak for any A∈Mn,q(K).Similarly, letL=F be a fuzzy semiring in Theorem 15. Then, we will also have the following.Corollary 18.
LetF be a fuzzy semiring. For any A∈Mn,q(F), denote ΦA=L- and JΦA={0,p1,p2,…,ph}. Assume that Ci(i=1,2,…,m) and ψ are given as in Lemma 1 Define the mapping φ from Mn,q to ∏i=1m‍Mn,q(Ci) by
(29)(∀A=(aij)∈Mn,q(L-))Aφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ. Then, φ is a subdirect embedding from semigroup Mn,q(L-) to semigroup ∏i=1m‍Mn,q(Ci) and A=∑k=1m‍Ak for any A∈Mn,q(F).Remark 19.
Theorem15 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 Section2 to give a way to calculate indices and periods of the square matrices over a distributive lattice.LetL be a distributive lattice and let Mn(L) be a semiring of matrices over L. Recall that if there exist positive integers d and p satisfying Ad+p=Ad for any (but fixed) A∈Mn(L), then the least such positive integers k and p are called the index and the period of A, respectively, and denoted by k(A) and p(A), respectively.Now, by Theorem7, it is not hard to obtain the following proposition.Theorem 20.
Assume thatψ is a subdirect embedding from distributive lattice L to the direct product ∏i=1m‍Li of distributive lattices L1,L2,…,Lm. Define the mapping φ from Mn(L) to ∏i=1m‍Mn(Li) by
(30)(∀A=(aij)∈Mn(L))Aφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ. Then, for any A∈Mn(L) and any positive integers k and p, one has
(31)Ad+p=Ad⟺(∀1≤k≤m)Akd+p=Akd.Proof.
For anyA∈Mn(L), assume that φ is a subdirect embedding from Mn(L) to ∏i=1m‍Mn(Li) with Aφ=(A1,A2,…,Am). If Ad+p=Ad, then we have
(32)Ad+pφ=Adφ;
that is,
(33)(A1,A2,…,Am)d+p=(A1,A2,…,Am)d,
and then
(34)(A1d+p,A2d+p,…,Amd+p)=(A1d,A2d,…,Amd).
Thus, Aid+p=Aid for i=1,2,…,m.
Conversely, assume thatAφ=(A1,A2,…,Am), and, for i=1,2,…,m, Aid+p=Aid. Then, we have
(35)(A1d+p,A2d+p,…,Amd+p)=(A1,A2,…,Am);
that is,
(36)(A1,A2,…,Am)d+p=(A1,A2,…,Ah)d,
and then
(37)Ad+pφ=Adφ.
Notice that φ is the subdirect embedding from Mn(L) to ∏i=1m‍Mn(Li); we immediately get Ad+p=Ad.It is not hard to see that the indices and periods of the square matrices over a distributive lattice must exist. By Lemma14, we can also get the following.Theorem 21.
LetL be a distributive lattice. For any A∈Mn(L), denote 〈ΦA〉=L-, J〈ΦA〉={0,p1,p2,…,ph}. Assume that Ci(i=1,2,…,m) and ψ are given as in Lemma 1. Define the mapping φ from Mn(L-) to ∏i=1m‍Mn(Ci) by
(38)(∀A=(aij)∈Mn(L-))Aφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ. Then, the index k(A) of A is equal to the maximum in the set of all indices of Ak(1≤k≤m) and the period p(A) of A is equal to the least common multiple of all periods of Ak(1≤k≤m).In Theorem21, take Ci={0,pi}; then, we have the following.Corollary 22.
LetL be a distributive lattice. For any A∈Mn(L), denote 〈ΦA〉=L-, J〈ΦA〉={0,p1,p2,…,ph}. Then, the index k(A) of A is equal to the maximum in the set of all indices of Api(1≤i≤h) and the period p(A) of A is equal to the least common multiple of all periods of Api(1≤i≤h).Remark 23.
Theorem21 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, considerL to be a finite distributive lattice in Theorem 21. By Corollary 22, we immediately have the following.Theorem 24.
LetL be a finite distributive lattice. For any A∈Mn(L), denote 〈ΦA〉=L-, J〈ΦA〉={0,p1,p2,…,ph}. Assume that Ci(i=1,2,…,m) and ψ are given as in Lemma 1. Define the mapping φ from Mn(L-) to ∏i=1m‍Mn(Ci) by
(39)(∀A=(aij)∈Mn(L-))Aφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ. Then, the index k(A) of A is equal to the maximum in the set of all indices of Ak(1≤k≤m) and the period p(A) of A is equal to the least common multiple of all periods of Ak(1≤k≤m).Corollary 25.
LetL be a finite distributive lattice. For any A∈Mn(L), denote 〈ΦA〉=L-, J〈ΦA〉={0,p1,p2,…,ph}. Then, the index k(A) of A is equal to the maximum in the set of all indices of Api(1≤i≤h) and the period p(A) of A is equal to the least common multiple of all periods of Api(1≤i≤h).Example 26.
LetL={0,a,b,c,d,…} be an infinite distributive lattice whose Hasse diagram is shown as in Example 16.
Now, for givenA=(caabb00ca)∈M3(L), then, 〈ΦA〉={0,a,b,c},J〈ΦA〉={0,a,b}.
Note by Lemma1 that finite distributive lattice {0,a,b,c} is isomorphic to a subdirect product of chains {0,a} and {0,b}, and then by Theorem 15 we have
(40)A=(caabb00ca)=(aaa0000aa)+(b00bb00b0)=A1+A2.
It is easy to check that k(A1)=1,k(A2)=2, and p(A1)=p(A2)=1. Hence, k(A)=2,p(A)=1.On the other hand, by Theorem20, we also have the following.Theorem 27.
LetL be a finite distributive lattice with l≥2 elements and J={0,p1,p2,…,ph} the set of all join irreducible elements of L. Assume that Ci(i=1,2,…,m) and ψ are given as in Lemma 1. Define the mapping φ from Mn(L) to ∏i=1m‍Mn(Ci) by
(41)(∀A=(aij)∈Mn(L))Aφ=(A1,A2,…,Am),
where (∀1≤k≤m)Ak=(aij(k)) and aij(k)=(k)(aij)ψ. Then, the index k(A) of A is equal to the maximum in the set of all indices of Ak(1≤k≤m) and the period p(A) of A is equal to the least common multiple of all periods of Ak(1≤k≤m).Similarly, if we takeCi={0,pi} in Theorem 27, then we will get the following theorem.Theorem 28.
LetL be a finite distributive lattice with l≥2 elements and A∈Mn(L)and J={0,p1,p2,…,ph} the set of all join irreducible elements of L. Then, the index k(A) of A is equal to the maximum in the set of all indices of Api(1≤i≤h) and the period p(A) of A is equal to the least common multiple of all periods of Api(1≤i≤h).By applying Theorem27 to a finite chain L={0,1,…,l}, together with the discussions of Corollary 13, we can also immediately obtain the following theorem which had been proved by [30].Theorem 29.
LetL={0,1,…,l}(l>1) be a finite chain with the usual ordering. For any A∈Mn(L), the index k(A) of A is equal to the maximum in the set of all indices of  φ-ls(k)(A)(1≤s≤m) and the period p(A) of A is equal to the least common multiple of all periods ofφ-ls(k)(A)(1≤s≤m).Next, using the decompositions of matrices obtained in Section2, 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.LetL be a distributive lattice and Mn(L) a semiring of matrices over L. Recall that a matrix A in Mn(L) is called idempotent if A2=A.Theorem 30.
LetL and Li(1≤i≤h) be distributive lattices. Assume that φ is a subdirect embedding from semiring Mn(L) to semiring ∏i=1h‍Mn(Li), and, for any A∈Mn(L), Aφ=(A1,A2,…,Ah). Then, A2=A if and only if Ai2=Ai(i=1,2,…,h).Proof.
For anyA∈Mn(L), assume that φ is a subdirect embedding from Mn(L) to ∏i=1h‍Mn(Li) with Aφ=(A1,A2,…,Ah). If A2=A, then we have
(42)A2φ=Aφ;
that is,
(43)(A1,A2,…,Ah)2=(A1,A2,…,Ah),
and then
(44)(A12,A22,…,Ah2)=(A1,A2,…,Ah).
Thus, Ai2=Ai for i=1,2,…,h.
Conversely, assume thatAφ=(A1,A2,…,Ah), and for i=1,2,…,h, Ai2=Ai. Then, we have
(45)(A12,A22,…,Ah2)=(A1,A2,…,Ah);
that is,
(46)(A1,A2,…,Ah)2=(A1,A2,…,Ah),
and then
(47)A2φ=Aφ.
Notice that φ is the subdirect embedding from Mn(L) to ∏i=1h‍Mn(Li); we immediately get A2=A.For anyA∈Mn(L), notice that 〈ΦA〉 is a finite distributive lattice; by Theorems 8 and 30, we immediately get the following.Theorem 31.
LetL be a distributive lattice. For any A∈Mn(L), denote 〈ΦA〉=L-. Assume that C-i(1≤i≤h) are the subdistributive lattices of L- given as in Lemma 1 and φ is a subdirect embedding from semiring Mn(L-) to semiring ∏i=1h‍Mn(C-i) with Aφ=(A1,A2,…,Ah). Then, A2=A if and only if Ai2=Ai  (i=1,2,…,h).By Theorem31, together with Theorem 8 or Corollary 10, we have the following.Corollary 32.
LetL be a distributive lattice. For any A∈Mn(L), A2=A if and only if Api2=Api(i=1,2,…,h), where 0≠pi∈J〈ΦA〉 and A=∑i=1h‍piApi.Example 33.
LetL={0,a,b,c,d,…} be an infinite distributive lattice whose Hasse diagram is shown as in Example 16.
Now, givenA=(caabb000a)∈M3(L), clearly,〈ΦA〉={0,a,b,c},J〈ΦA〉={0,a,b}.
Note by Lemma1 that finite distributive lattice {0,a,b,c} is isomorphic to a subdirect product of chains {0,a} and {0,b}, and then, by Theorem 15, we have
(48)A=(caabb000a)=a(111000001)+b(100110000).
By direct calculation (or by Theorem 2.11 in [14]), it is easy to check that
(49)(111000001),(100110000)areidempotentmatrices.
Thus, by Corollary 32,
(50)A=(caabb000a)isanidempotentmatrix.LetL=K be a chain semiring in Corollary 32. Then, for any A∈Mn(K), notice that 〈ΦA〉=ΦA for any A∈Mn(K); we have the following corollary.Corollary 34.
LetA be a matrix in Mn(K). Then, A is idempotent if and only if all Api are idempotent, where all pi are the nonzero join irreducible elements of ΦA and A=∑i=1h‍piApi.Similarly, letL=F be a fuzzy semiring in Corollary 32. Then, we will also have the following.Corollary 35.
LetA be a matrix in Mn(F). Then, A is idempotent if and only if all Api are idempotent, where all pi are the nonzero join irreducible elements of ΦA and A=∑i=1h‍piApi.Remark 36.
Theorem31 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 Theorem30, consider L to be a finite distributive lattice. By Lemma 3 and Corollary 10, we immediately have the following.Corollary 37.
LetL be a finite distributive lattice and pi(i=1,2,…,h) be its all nonzero join irreducible elements. Then, for any A∈Mn(L), A2=A if and only if Api2=Api(i=1,2,…,h), where A=∑i=1h‍piApi.In particular, if we letL=Bk 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.
LetA be a matrix in Mn(Bk) with k≥1. Then, A is idempotent if and only if all pth constituents of A are idempotent in Mn(B1), where the pth constituent of A is the matrix in Mn(B1) whose (i,j)th entry is 1 if and only if aij⊇σp.On the other hand, by Theorem31, we will also have the following results.Corollary 39.
LetL be a finite distributive lattice. For any A∈Mn(L), denote 〈ΦA〉=L-. Assume that L-i(1≤i≤h) are the subdistributive lattices of L- and φ is a subdirect embedding from semiring Mn(L-) to semiring ∏i=1h‍Mn(L-i) with Aφ=(A1,A2,…,Ah). Then, A2=A if and only if Ai2=Ai(i=1,2,…,h).Corollary 40.
LetL be a finite distributive lattice. For any A∈Mn(L), A2=A if and only if Api2=Api(i=1,2,…,h), where 0≠pi∈J〈ΦA〉 and A=∑i=1h‍piApi.Finally, we will use the decompositions of matrices obtained in Section2 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.LetL be a distributive lattice and let Mn(L) be a semiring of matrices over L. Recall that a matrix A in Mn(L) is called nilpotent if Ak=O for some k≥1, where O is the zero matrix in Mn(L).Theorem 41.
LetL and Li(1≤i≤h) be distributive lattices. Assume that φ is a subdirect embedding from semiring Mn(L) to semiring ∏i=1h‍Mn(Li), and, for any A∈Mn(L), Aφ=(A1,A2,…,Ah). Then, Ak=O if and only if Aik=O(i=1,2,…,h).Proof.
Assume thatφ is a subdirect embedding from Mn(L) to ∏i=1h‍Mn(Li) and, for any A∈Mn(L), Aφ=(A1,A2,…,Ah). If An=O, then we have Anφ=O; that is,
(51)(A1,A2,…,Ah)n=(O,O,…,O),
and then
(52)(A1n,A2n,…,Ahn)=(O,O,…,O).
Thus, we have Ain=O for i=1,2,…,h.
Conversely, fori=1,2,…,h, if Ain=O, then we have Anφ=(O,O,…,O). Notice that φ is the subdirect embedding from Mn(L) to ∏i=1h‍Mn(Li); then, we immediately have An=O.By Theorems8 and 41, we can get the following.Theorem 42.
LetL be a distributive lattice. For any A∈Mn(L), denote 〈ΦA〉=L-. Assume that C-i(1≤i≤h) are the subdistributive lattices of L- given as in Lemma 1 and φ is a subdirect embedding from semiring Mn(L-) to semiring ∏i=1h‍Mn(C-i) with Aφ=(A1,A2,…,Ah). Then, Ak=O if and only if Aik=O(i=1,2,…,h).By Theorem42, together with Theorem 8 or Corollary 10, we have the following.Corollary 43.
LetL be a distributive lattice. For any A∈Mn(L), Ak=O if and only if Apik=O(i=1,2,…,h), where 0≠pi∈J〈ΦA〉 and A=∑i=1h‍piApi.Example 44.
LetL={0,a,b,c,d,…} be an infinite distributive lattice whose Hasse diagram is shown as in Example 16.
For givenA=(caabb000a)∈M3(L), clearly,〈ΦA〉={0,a,b,c},J〈ΦA〉={0,a,b}.
Note by Lemma1 that finite distributive lattice {0,a,b,c} is isomorphic to a subdirect product of chains {0,a} and {0,b}, and then, by Theorem 15, we have
(53)A=(0ab00c000)=(0a000a000)+(00b00b000).
By direct calculation, it is easy to check that
(54)(010001000),(001001000)arenilpotentmatrices.
And then by Corollary 43,
(55)A=(0ab00c000)isanilpotentmatrix.LetL=K be a chain semiring in Corollary 43. Then, for any A∈Mn(K), notice that 〈ΦA〉=ΦA for any A∈Mn(K); we have the following corollary.Corollary 45.
LetA be a matrix in Mn(K). Then, A is nilpotent if and only if all Api are nilpotent, where all pi are the nonzero join irreducible elements of ΦA and A=∑i=1h‍piApi.Similarly, letL=F be a fuzzy semiring in Corollary 43. Then, we will also have the following.Corollary 46.
LetA be a matrix in Mn(F). Then, A is nilpotent if and only if all Api are nilpotent, where all pi are the nonzero join irreducible elements of ΦA and A=∑i=1h‍piApi.Remark 47.
Theorem42 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 Theorem41, consider L to be a finite distributive lattice. By Lemma 3 and Corollary 10, we immediately have the following.Corollary 48.
LetL be a finite distributive lattice and let pi(i=1,2,…,h) be its all nonzero join irreducible elements. Then, for any A∈Mn(L), An=O if and only if Apin=O(i=1,2,…,h), where A=∑i=1h‍piApi.In particular, if we letL=Bk be the general Boolean algebra in Corollary 48, then we will obtain the following corollary.Corollary 49.
LetA be a matrix in Mn(Bk) with k≥1. Then, A is nilpotent if and only if all pth constituents of A are nilpotent in Mn(B1), where the pth constituent of A is the matrix in Mn(B1) whose (i,j)th entry is 1 if and only if aij⊇σp.On the other hand, by Theorem42, we will also obtain the following corollaries.Corollary 50.
LetL be a finite distributive lattice. For any A∈Mn(L), denote 〈ΦA〉=L-. Assume that L-i(1≤i≤h) are the subdistributive lattices of L- and φ is a subdirect embedding from semiring Mn(L-) to semiring ∏i=1h‍Mn(L-i) with Aφ=(A1,A2,…,Ah). Ak=O if and only if Aik=O(i=1,2,…,h).Corollary 51.
LetL be a finite distributive lattice. For any A∈Mn(L), Ak=O if and only if Apik=O(i=1,2,…,h), where 0≠pi∈J〈ΦA〉 and A=∑i=1h‍piApi.



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