On Subdirect Decompositions of Finite Distributive Lattices

Hindawi Discrete Dynamics in Nature and Society Volume 2017, Article ID 6490903, 5 pages https://doi.org/10.1155/2017/6490903

Research Article On Subdirect Decompositions of Finite Distributive Lattices

Yizhi Chen,1 Jing Tian,2 and Zhongzhu Liu1

1 School of Mathematics and Big Data, Huizhou University, Huizhou, Guangdong 516007, China 2School of Economics and Finance, Xi’an International Studies University, Xi’an, Shaanxi 710128, China

Correspondence should be addressed to Yizhi Chen; [email protected]

Received 18 January 2017; Accepted 2 April 2017; Published 27 April 2017

AcademicEditor:J.R.Torregrosa

Copyright © 2017 Yizhi Chen et al. 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.

Subdirect decomposition of algebra is one of its quite general and important constructions. In this paper, some subdirect decompositions (including subdirect irreducible decompositions) of finite distributive lattices and finite chains are studied, and some general results are obtained.

1. Introduction and Preliminaries An embedding 𝜙:𝑅→Π𝑖∈𝐼𝑅𝑖 is subdirect if 𝑅𝜙 is asubdirectproductof𝑅𝑖.Atthistime,wealsosaythat𝑅 𝑅 (𝑅,+,⋅) A semiring is an algebraic structure consisting of has subdirect decomposition of 𝑅𝑖 or 𝑅 is isomorphic to the 𝑅 a nonempty set together with two binary operations + and subdirect product of {𝑅𝑖}𝑖∈𝐼. ⋅ 𝑅 (𝑅, +) (𝑅, ⋅) on such that and are semigroups connected A semiring 𝑅 is called subdirectly irreducible if for every 𝑎(𝑏 + 𝑐) = 𝑎𝑏 +𝑎𝑐 (𝑏 + 𝑐)𝑎 = by distributivity, that is, and subdirect embedding 𝜙:𝑅→Π𝑖∈𝐼𝑅𝑖,thereisan𝑖∈𝐼such 𝑏𝑎+𝑐𝑎 𝑎, 𝑏, 𝑐∈𝑅 𝑅 ,forall [1, 2]. A semiring is called a partially that 𝜙∘𝜋𝑖 :𝑅→𝑅𝑖 is an isomorphism. From the above ≤ ordered semiring if it admits a compatible ordering ,thatis, definition, it is easy to see that any two-element semiring is ≤ 𝑅 is a partial order on satisfying the following condition: subdirectly irreducible. 𝑎, 𝑏, 𝑐, 𝑑∈𝑅 𝑎≤𝑏 𝑐≤𝑑 𝑎+𝑐 ≤ 𝑏+𝑑 for any ,if and ,then and From [4–7] we know that the subdirect product is a quite 𝑎𝑐 ≤ 𝑏𝑑 𝑅 . A partially ordered semiring is said to be a totally general construction. As for as semirings concerned, there ordered semiring if the imposed partial order is a total order are several ways of approaching subdirect decompositions of [1, 2]. semirings. In most cases they can be obtained from various A distributive lattice is a lattice which satisfies the dis- semirings theoretical constructions. Another way is based 𝐿= tributive laws [3]. In the following, we will denote on the famous Birkhoff representation theorem. Formulated ⟨𝐿,+,⋅,0,1⟩ 0 1 as a finite distributive lattice, where and are in terms of semirings, it asserts that every semiring can be 𝐿 theleastandthegreatestelementsof ,respectively,andthe represented as a subdirect product of subdirectly irreducible 𝐿 addition and the multiplication on are defined as follows: semirings, and it can often reduce studying the structure 𝑎+𝑏=𝑎∨𝑏=max {𝑎,} 𝑏 , of semirings from a given class to studying subdirectly irreducible members of this class. Also, there is a third way 𝑎⋅𝑏=𝑎∧𝑏=min {𝑎,} 𝑏 (1) of approaching subdirect decompositions which is based on ∀𝑎, 𝑏 ∈ 𝐿. another Birkhoff theorem verified in [4], which, in terms of semirings, says that a semiring 𝑅 isasubdirectproductofa Also, we denote 𝐿 = {0,1,...,𝑚}as a finite chain with usual family of semirings {𝑅𝑖}𝑖∈𝐼 if and only if there exists a family ordering [4]. Clearly, both the finite distributive lattices and of factor congruences {𝜌𝑖}𝑖∈𝐼 on 𝑅 such that ⋂𝑖∈𝐼 𝜌𝑖 =𝜀𝑅 and the finite chains are partially ordered semirings. 𝑅/𝜌𝑖 =𝑅𝑖 for each 𝑖∈𝐼;here,𝜀𝑅 is the identity congruence A semiring 𝑅 issaidtobeasubdirectproductofan on 𝑅. indexed family 𝑅𝑖 (𝑖 ∈ 𝐼) ofsemiringsifitsatisfies𝑅≤Π𝑖∈𝐼𝑅𝑖 The main aim of this paper is to investigate the subdirect and 𝑅𝜋𝑖 =𝑅𝑖 for each 𝑖∈𝐼. decompositions of a special class of semirings called finite 2 Discrete Dynamics in Nature and Society

𝑚 distributive lattices. Although some subdirect decomposi- and 𝑝≤⋁𝑖=1𝑥𝑖 (𝑥𝑖 ∈𝐽,1≤𝑖≤𝑚), then there exists 𝑖0, 1≤𝑖 ≤𝑚 𝑝≤𝑥 . tions of a finite distributive lattice are discussed in [8], and 0 ,suchthat 𝑖0 the subdirect decompositions of a finite chain are studied in [2], the results in this paper will be more general. We By Lemmas 6 and 7, we immediately obtain the following will investigate some subdirect decompositions (including corollary. subdirect irreducible decompositions) of finite distributive Corollary 8. 𝐿 𝐽 lattices and finite chains, whose proofs are also different from Let be a finite distributive lattice and be the set of all the join irreducible elements of 𝐿.If𝑝∈𝐽and 𝑝≤ [8]. 𝑘 For notations and terminologies occurred but not men- ⋁𝑖=1𝑥𝑖 (𝑥𝑖 ∈𝐿,1≤𝑖≤𝑘), then there exists 𝑖0, 1≤𝑖0 ≤𝑘, 𝑝≤𝑥 tioned in this paper, the readers are referred to [1, 4]. such that 𝑖0 .

Now, we can give some subdirect irreducible decomposi- 2. Subdirect Decompositions of tions of a finite distributive lattice 𝐿. Finite Distributive Lattices Theorem 9. Let 𝐿 be a finite distributive lattice with 𝑚+1 To obtain our main results in this section, we will also need elements where 𝑚≥1and 𝐽={0,𝑝1,𝑝2,...,𝑝ℎ}(𝑚≥ℎ≥1) the following lemmas and concepts. are the set of all the join irreducible elements of 𝐿.Then𝐿 is isomorphic to a subdirect product of subdirect irreducible Lemma 1 (see [4]). In the equational class of distributive elements 𝐿𝑖 ={0,𝑝𝑖} (𝑖=1,2,...,ℎ). lattices the only nontrivial subdirectly irreducible algebra is the two-element chain. Proof. For 𝑖=1,2,...,ℎ, define 𝜙𝑖 :𝐿→𝐿𝑖 by 𝑅 𝑅 Let be a semiring and Con be the set of all congruences {𝑝 ,𝑎≥𝑝, on 𝑅.ByLemma8.2in[4],wehavethefollowinglemma. 𝑖 𝑖 𝑎𝜙𝑖 = { (4) 0, . 𝑚 { otherwise Lemma 2. If 𝜌𝑖 ∈ Con𝑅 for 𝑖∈𝐼and ∩𝑖=1𝜌𝑖 =𝜀𝑅, then the natural homomorphism Then, it is a routine way to verify that 𝜙𝑖 (𝑖= 1,2,...,ℎ)is a homomorphism. 𝛼:𝑅→Π𝑖∈𝐼𝑅𝑖 (2) Firstly, 𝜙𝑖 (𝑖=1,2,...,ℎ)is clearly a mapping. Secondly, for any 𝑎, 𝑏,wewillshowthat ∈𝐿 (𝑎 ∨ 𝑏)𝜙𝑖 = defined by 𝑎𝜙𝑖 ∨𝑏𝜙𝑖 (𝑖=1,2,...,ℎ). 𝑎 (𝑖)(𝑎) 𝛼= (i) If 𝑎∨𝑏≥𝑝𝑖,then,byCorollary8,𝑎≥𝑝𝑖 or 𝑏≥𝑝𝑖, 𝜌 (3) 𝑖 andthenweget(𝑎 ∨ 𝑏)𝜙𝑖 =𝑝𝑖 =𝑎𝜙𝑖 ∨𝑏𝜙𝑖. 𝑎∨𝑏≥𝑝̸ 𝑎≥𝑝 ̸ 𝑏≥𝑝 ̸ (𝑎 ∨ is a subdirect embedding. (ii) If 𝑖,then 𝑖 and 𝑖,andthen 𝑏)𝜙𝑖 =0=0∨0=𝑎𝜙𝑖 ∨𝑏𝜙𝑖. In the following, we will discuss the subdirect decompo- (𝑎∧𝑏)𝜙 =𝑎𝜙∧𝑏𝜙 (𝑖=1,2,...,ℎ) sitions of a finite distributive lattice. Thirdly, we show that 𝑖 𝑖 𝑖 for any 𝑎, 𝑏. ∈𝐿 Definition 3 (see [4]). Let 𝐿 be a lattice. The element 𝑎∈𝐿is (i) If 𝑎∧𝑏≥𝑝𝑖,then𝑎≥𝑝𝑖 and 𝑏≥𝑝𝑖.Thus,wehave called join irreducible of 𝐿 if for 𝑥, 𝑦, ∈𝐿 𝑎=𝑥∨𝑦implies (𝑎 ∧ 𝑏)𝜙𝑖 =𝑝𝑖 =𝑝𝑖 ∧𝑝𝑖 =𝑎𝜙𝑖 ∧𝑏𝜙𝑖. 𝑥=𝑎or 𝑦=𝑎. (ii) If 𝑎∧𝑏 ≥𝑝̸ 𝑖,then𝑎≥𝑝 ̸ 𝑖 or 𝑏≥𝑝𝑖,andthen(𝑎∧𝑏)𝜙𝑖 = 0=𝑎𝜙 ∧𝑏𝜙. Example 4. Let 𝐿36 denote the divisible lattice which is 𝑖 𝑖 generated by all the positive factor of 36; then the set 𝐽 of all Summing up all the discussions above, we have shown the join irreducible of 𝐿36 is {1,2,3,4,9}. that 𝜙𝑖 (𝑖=1,2,...,ℎ)is a homomorphism. 𝜌 = 𝜙 𝜌=∩ℎ 𝜌 =𝜀 Definition 5 (see [4]). Let 𝐿 be a finite distributive lattice Now, let 𝑖 Ker 𝑖. Next, we show that 𝑖=1 𝑖 𝐿. 𝑎, 𝑏 ∈𝐿 (𝑎, 𝑏) ∈𝜌 and 𝑎∈𝐿. If there exist join irreducible elements 𝑥𝑖 (𝑖 = Assume that and ,thenwehave 𝑠 (𝑎, 𝑏) ∈𝜌 (𝑖 = 1,2,...,ℎ) 1,2,...,𝑠)such that 𝑎=⋁𝑖=1𝑥𝑖,where𝑥𝑖 ⩽𝑥¡ 𝑗 for any 1≤𝑖 ̸= 𝑖 . By Lemma 6, there exist the 𝑠 join irreducible decompositions of 𝑎 and 𝑏. Assume that 𝑗≤𝑠,thenwecall⋁𝑖=1𝑥𝑖 the join irreducible decomposition 𝑎=𝑝𝑗 ⋁𝑝𝑗 ⋁⋅⋅⋅⋁𝑝𝑗 and 𝑏=𝑝𝑖 ⋁𝑝𝑖 ⋁⋅⋅⋅⋁𝑝𝑖 , of 𝑎. 1 2 𝑠 1 2 𝑡 where 𝑝𝑗 ,𝑝𝑗 ,...,𝑝𝑗 ,𝑝𝑖 ,𝑝𝑖 ,...,𝑝𝑖 ∈𝐽.Thenweget𝑎≥ 𝑝 ,𝑝 ,...,𝑝1 2 𝑠𝑏≥𝑝1 2 ,𝑝 ,...,𝑝𝑡 (𝑎, 𝑏) ∈ Lemma 6 (see [4]). Let 𝐿 be a finite distributive lattice and 𝑗1 𝑗2 𝑗𝑠 and 𝑖1 𝑖2 𝑖𝑡 .Since 𝜌 (𝑖 = 1,2,...,ℎ) 𝑏≥𝑝,𝑝 ,...,𝑝 𝑎≥ 𝐽 be the set of all the join irreducible elements of 𝐿.Thenfor 𝑖 ,wehave 𝑗1 𝑗2 𝑗𝑠 , 𝑎∈𝐿,𝑎 𝑝 ,𝑝 ,...,𝑝 𝑎=𝑝 ⋁𝑝 ⋁⋅⋅⋅⋁𝑝 ≤𝑏𝑏= any has a unique join irreducible decomposition and 𝑖1 𝑖2 𝑖𝑡 ,andthen 𝑗1 𝑗2 𝑗𝑠 , 𝑎=⋁ 𝑝 𝑝 ⋁𝑝 ⋁⋅⋅⋅⋁𝑝 ≤𝑎 𝑎=𝑏 𝑝∈𝐽,𝑝≤𝑎 . 𝑖1 𝑖2 𝑖𝑡 .Hence,weobtain . Now, by Lemmas 1 and 2, we immediately verify that 𝐿 Lemma 7 (see [4]). Let 𝐿 be a finite distributive lattice and is isomorphic to a subdirect product of subdirect irreducible 𝐽 be the set of all the join irreducible elements of 𝐿.If𝑝∈𝐽 elements 𝐿𝑖 (𝑖=1,2,...,ℎ). Discrete Dynamics in Nature and Society 3

1 1

c b

Figure 2 0

Figure 1 1

Example 10. Let 𝐿={0,𝑎,𝑏,𝑐,1}be a finite distributive lattice d as the Hasse diagram shown in Figure 1. Clearly, 𝐽={0,𝑎,𝑏,1}.Now,wecantake𝐿1 ={0,𝑎}, 𝐿2 = b c {0, 𝑏},and𝐿3 = {0, 1} and define

3 a 𝜙:𝐿󳨀→∏𝐿𝑖, 𝑖=1 0 0 󳨃󳨀→ (0, 0, 0) , Figure 3 𝑎 󳨃󳨀→ (𝑎, 0, 0) , (5) 𝑏 󳨃󳨀→ (0, 𝑏, 0) ,

satisfying 0=𝑝𝑖 <𝑝𝑖 <⋅⋅⋅<𝑝𝑖 , 𝑠1 +𝑠2 +⋅⋅⋅+𝑠𝑟 =ℎ,and 𝑐 󳨃󳨀→ (𝑎, 𝑏,) 0 , 0 1 𝑠𝑖 𝐿𝑖 ⋂ 𝐿𝑘 ={0}if 𝑖=𝑘 ̸ .Notethatthemaximalchainfrom0toa 1 󳨃󳨀→ (𝑎, 𝑏,) 1 . maximal element of 𝐽 may be not unique, and the expression 𝑟 ⋃𝑖=𝑙 𝐿𝑖 may be not unique in general. Then it is a routine way to check that 𝜙 is a subdirect 𝑙 3 Further, we can take 𝐿𝑖 = ⋃𝑗=𝑙 𝐿𝑖𝑗 (1≤𝑖≤𝑟), where embedding homomorphism from 𝐿 to ∏𝑖=1𝐿𝑖.Hence,𝐿 is 𝐿𝑖𝑗 ={0,𝑝𝑖𝑗 ,...,𝑝𝑖𝑗 } is a more than 2 elements subchain isomorphic to a subdirect product of subdirect irreducible 1 𝑡𝑗 𝐿 (𝑖 = 1, 2, 3) 𝐿 𝐿 0=𝑝 < elements 𝑖 . of 𝑖 (and also a subchain of )andsatisfying 𝑖𝑗0 𝑝𝑖𝑗 < ⋅⋅⋅ < 𝑝𝑖𝑗 , 𝑡1 +𝑡2 +⋅⋅⋅+𝑡𝑙 =𝑠𝑖,and𝐿𝑖𝑗 ⋂ 𝐿𝑖𝑘 ={0} In general, if we replace the finite distributive lattice 1 𝑡𝑗 𝑗=𝑘 ̸ 𝐽=⋃𝑟 𝐿 =⋃𝑟 ⋃𝑙 𝐿 𝐿 with a finite lattice, we cannot get the corresponding if . Clearly, we have 𝑖=𝑙 𝑖 𝑖=𝑙 𝑗=𝑙 𝑖𝑗 ,where 𝐿 𝐿𝑖𝑗 ={0,𝑝𝑖𝑗 ,...,𝑝𝑖𝑗 } is a subchain of 𝐿 and constructed as subdirect irreducible decomposition of . 1 𝑡𝑗 above. Example 11. Let 𝐿={0,𝑎,𝑏,𝑐,1}be a finite lattice as the Hasse diagram shown in Figure 2. Example 12. Let 𝐿 be a finite distributive lattice whose Hasse 𝐽 = {0,𝑎,𝑏,𝑐} 𝐿 ={0,𝑎} Clearly, .Now,ifwetake 1 , diagram given as shown in Figure 3. Clearly, 𝐽={0,𝑎,𝑏,𝑐,1}, 𝐿2 ={0,𝑏},and𝐿3 ={0,𝑐}, then there is not any existing and we can take 𝐽=𝐿1 ⋃ 𝐿2,where𝐿1 = {0, 𝑎, 𝑏, 1} and 𝜙 𝐿 ∏3 𝐿 subdirect embedding homomorphism from to 𝑖=1 𝑖. 𝐿2 ={0,𝑐}(also, we can take 𝐿1 = {0, 𝑎, 𝑐, 1} and 𝐿2 ={0,𝑏}). 𝐿 Hence, is not isomorphic to a subdirect product of subdirect If we take 𝐿11 = {0, 𝑎, ,𝑏} 𝐿12 = {0, 1},and𝐿21 ={0,𝑐}, 𝐿 (𝑖=1,2,3) irreducible elements 𝑖 . then 𝐽=𝐿11 ⋃𝐿12 ⋃𝐿21.Ifwetake𝐿11 ={0,𝑎}, 𝐿12 = {0, 𝑏}, 𝐿13 = {0, 1},and𝐿21 ={0,𝑐},thenwecanget𝐽= Next, we will discuss more general subdirect decomposi- 𝐿11 ⋃𝐿12 ⋃𝐿13 ⋃𝐿21. tions of a finite distributive lattice. 𝐿 𝑚+1 Let be a finite distributive lattice with elements Theorem 13. Let 𝐿 be a finite distributive lattice with 𝑚+1 𝑚≥1 𝐽={0,𝑝,𝑝 ,...,𝑝 }(𝑚≥ ℎ ≥ 1) where and 1 2 ℎ are elements where 𝑚≥1and 𝐽={0,𝑝1,𝑝2,...,𝑝ℎ}(𝑚≥ℎ≥1) the set of all the join irreducible elements of 𝐿.Then𝐽 can be 𝐿 𝐿 𝑟 are the set of all the join irreducible elements of .Then is expressed as ⋃𝑖=𝑙 𝐿𝑖;here𝐿𝑖 ={0,𝑝𝑖 ,...,𝑝𝑖 } is a more than 𝐿 (1≤𝑖≤𝑟,1≤𝑗≤𝑙) 1 𝑠𝑖 isomorphic to a subdirect product of 𝑖𝑗 2 elements maximal subchain of 𝐽 (and also a subchain of 𝐿) which is constructed as above. 4 Discrete Dynamics in Nature and Society

Proof. For 1≤𝑖≤𝑟, 1≤𝑗≤𝑙, define 𝜙𝑖𝑗 :𝐿→𝐿𝑖𝑗 by In general, the subdirect decomposition manners of a finite chain into the subdirect irreducible elements can be 𝑎𝜙𝑖𝑗 various. 𝐿 = {0,1,...,𝑚} {𝑝𝑖𝑗 ,𝑎≥𝑝𝑖𝑗 , (6) Example 15. Let be a finite chain. By the 𝑡𝑗 𝑡𝑗 = { abovecorollary,wehaveshownthat𝐿 is isomorphic to a 𝑝𝑖𝑗 ,𝑎≥𝑝𝑖𝑗 ,𝑎≥𝑝̸ 𝑖𝑗 (0 ≤ 𝑡𝑘 ≤𝑡𝑗 −1). 𝐿 { 𝑡𝑘 𝑡𝑘 𝑡𝑘+1 subdirect product of subdirect irreducible elements 𝑖,where 𝐿𝑖 = {0,𝑖}(𝑖= 1,2,...,𝑚). On the other hand, we can also 𝜙 (1≤𝑖≤𝑟,1≤ 󸀠 Then, it is a routine way to verify that 𝑖𝑗 take 𝐿𝑖 ={𝑖−1,𝑖}(𝑖=1,2,...,𝑚)and define 𝑗≤𝑙)is a homomorphism. 𝜙 (1≤𝑖≤𝑟,1≤𝑗≤𝑙) 𝑚 Firstly, 𝑖𝑗 is clearly a mapping. 𝜙󸀠 :𝐿→∏𝐿󸀠, 𝑎, 𝑏 ∈𝐿 (𝑎 ∨ 𝑏)𝜙 = 𝑖 Secondly, for any ,wewillshowthat 𝑖𝑗 𝑖=1 𝑎𝜙𝑖 ∨𝑏𝜙𝑖𝑗 (1≤𝑖≤𝑟,1≤𝑗≤𝑙). 0 󳨃󳨀→ (0,1,2,...,𝑚−2,𝑚−1) , (i) If 𝑎∨𝑏≥𝑝𝑖𝑗 ,then,byCorollary8,𝑎≥𝑝𝑖𝑗 or 𝑏≥ 𝑡𝑗 𝑡𝑗 1 󳨃󳨀→ (1,1,2,...,𝑚−2,𝑚−1) , 𝑝𝑖𝑗 ,andthenweget(𝑎 ∨ 𝑏)𝜙𝑖𝑗 =𝑝𝑖𝑗 =𝑎𝜙𝑖𝑗 ∨𝑏𝜙𝑖𝑗 . 𝑡𝑗 𝑡𝑗 2 󳨃󳨀→ (1,2,2,...,𝑚−2,𝑚−1) , (ii) If 𝑎∨𝑏≥𝑝𝑖𝑗 𝑎∨𝑏≥𝑝̸ 𝑖𝑗 (0 ≤ 𝑡𝑘 ≤𝑡𝑗 −1),then, 𝑡𝑘 𝑡𝑘+1 (7) by Corollary 8, 𝑎≥𝑝𝑖𝑗 or 𝑏≥𝑝𝑖𝑗 ,andalsowehave 3 󳨃󳨀→ (1,2,3,...,𝑚−2,𝑚−1) , 𝑡𝑘 𝑡𝑘 𝑎≥𝑝 ̸ 𝑏≥𝑝 ̸ (𝑎 ∨ 𝑏)𝜙 =𝑝 = 𝑖𝑗𝑡 +1 and 𝑖𝑗𝑡 +1 .Thus, 𝑖 𝑖𝑗𝑡 𝑘 𝑘 𝑘 . 𝑎𝜙𝑖𝑗 ∨𝑏𝜙𝑖𝑗 . .

Thirdly, we show that (𝑎 ∧ 𝑏)𝜙𝑖𝑗 =𝑎𝜙𝑖𝑗 ∧𝑏𝜙𝑖𝑗 (1 ≤ 𝑖 ≤ 𝑚−1󳨃󳨀→ (1,2,3,...,𝑚−1,𝑚−1) , 𝑟, 1 ≤ 𝑗 ≤ 𝑙) for any 𝑎, 𝑏. ∈𝐿 𝑚 󳨃󳨀→ (1,2,3,...,𝑚−1,𝑚) ; (i) If 𝑎∧𝑏≥𝑝𝑖𝑗 ,then𝑎≥𝑝𝑖𝑗 and 𝑏≥𝑝𝑖𝑗 .Thus,we 𝑡𝑗 𝑡𝑗 𝑡𝑗 󸀠 then it is not hard to check that 𝜙 is also a subdirect have (𝑎 ∧ 𝑏)𝜙𝑖𝑗 =𝑝𝑖𝑗 =𝑝𝑖𝑗 ∧𝑝𝑖𝑗 =𝑎𝜙𝑖𝑗 ∧𝑏𝜙𝑖𝑗 . 𝑚 𝑡𝑗 𝑡𝑗 𝑡𝑗 󸀠 embedding homomorphism from 𝐿 to ∏𝑖=1𝐿𝑖.Andso𝐿 is (ii) If 𝑎∧𝑏≥𝑝𝑖𝑗 , 𝑎∧𝑏≥𝑝̸ 𝑖𝑗 (0 ≤ 𝑡𝑘 ≤𝑡𝑗 −1),then isomorphic to a subdirect product of subdirect irreducible 𝑡𝑘 𝑡𝑘+1 󸀠 𝑎≥𝑝 𝑏≥𝑝 𝑎≥𝑝 ̸ elements 𝐿𝑖 (𝑖= 1,2,...,𝑚).Obviously,thetwomannersof we have 𝑖𝑗𝑡 and 𝑖𝑗𝑡 ,andalso 𝑖𝑗𝑡 or 𝑘 𝑘 𝑘 subdirect decomposition of 𝐿 into the subdirect irreducible 𝑏≥𝑝𝑖𝑗 .Hence,(𝑎 ∧ 𝑏)𝜙𝑖𝑗 =𝑝𝑖𝑗 =𝑎𝜙𝑖𝑗 ∧𝑏𝜙𝑖𝑗 . 𝑡𝑘 𝑡𝑘 elements are different. Summing up all the discussions above, we have shown 𝜙 (1≤𝑖≤𝑟,1≤𝑗≤𝑙) Next, we will denote ℎ = ⌈𝑚/2⌉,whichistheleastinteger that 𝑖𝑗 is a homomorphism. 𝑚/2 𝐿 = {0, 2𝑖 − 1, 2𝑖} (𝑖= 𝜌 = 𝜙 𝜌=∩ = greaterthanorequalto ,andtake 𝑖 Now, let 𝑖𝑗 Ker 𝑖𝑗 .Next,weshowthat 1≤𝑖≤𝑟,1≤𝑗≤𝑙 1,2,...,ℎ−1) 𝐿 = {0, 2ℎ − 1, 2ℎ} 𝑚=2ℎ 𝐿 = 𝜀 and ℎ ,when , ℎ 𝐿. {0, 2ℎ − 1} 𝑚=2ℎ−1 Assume that 𝑎, 𝑏 ∈𝐿 and (𝑎, 𝑏) ∈𝜌;thenwehave when . Also, we have the following theorem. (𝑎, 𝑏)𝑖𝑗 ∈𝜙 (1≤𝑖≤𝑟,1≤𝑗≤𝑙). By Lemma 6, there 𝑎 𝑏 exist the join irreducible decompositions of and . Assume Theorem 16. 𝐿 = {0,1,...,𝑚} 𝑎=𝑝 ⋁ 𝑝 ⋁ ⋅⋅⋅⋁ 𝑝 𝑏=𝑝 ⋁ 𝑝 ⋁ ⋅⋅⋅⋁ 𝑝 Let be a finite chain. Then that 𝜆 𝜆 𝜆 󸀠 and 𝜇 𝜇 𝜇 󸀠 , 1 2 𝑠 1 2 𝑡 𝐿 is isomorphic to a subdirect product of 𝐿𝑖 (𝑖 = 1,2,...,ℎ) where 𝑝𝜆 ,𝑝𝜆 ,...,𝑝𝜆 ,𝑝𝜇 ,𝑝𝜇 ,...,𝑝𝜇 ∈𝐽.Thenweget 1 2 𝑠󸀠 1 2 𝑡󸀠 constructed above. 𝑎≥𝑝 ,𝑝 ,...,𝑝 𝑏≥𝑝 ,𝑝 ,...,𝑝 (𝑎, 𝑏) ∈ 𝜆1 𝜆2 𝜆 󸀠 and 𝜇1 𝜇2 𝜇 󸀠 .Since 𝑠 𝑡 𝑖=1,2,...,ℎ 𝜙 :𝐿→𝐿 𝜌𝑖𝑗 (1≤𝑖≤𝑟,1≤𝑗≤𝑙) 𝑏≥𝑝𝜆 ,𝑝𝜆 ,...,𝑝𝜆 Proof. For , define 𝑖 𝑖 by ,wehave 1 2 𝑠󸀠 and 𝑎≥𝑝 ,𝑝 ,...,𝑝 𝑎=𝑝 ⋁ 𝑝 ⋁ ⋅⋅⋅⋁ 𝑝 ≤𝑏 𝜇1 𝜇2 𝜇 󸀠 and then 𝜆1 𝜆2 𝜆 󸀠 2𝑖, 𝑎 ≥ 2𝑖, 𝑡 𝑠 { 𝑏=𝑝𝜇 ⋁ 𝑝𝜇 ⋁ ⋅⋅⋅⋁ 𝑝𝜇 ≤𝑎 𝑎=𝑏 { and 1 2 𝑡󸀠 .Hence,weobtain . 𝐿 𝑎𝜙𝑖 = {2𝑖−1, 2𝑖>𝑎≥2𝑖−1, (8) Now, by Lemma 2, we immediately verify that is { 𝐿 (1≤𝑖≤𝑟,1≤ isomorphic to a subdirect product of 𝑖𝑗 {0, otherwise. 𝑗≤𝑙). Then, we can check that 𝜙𝑖 (𝑖 = 1,2,...,ℎ) is a homomor- In the following, we will discuss the subdirect product phism by a routine way. Now, let 𝜌𝑖 = Ker𝜙𝑖.Also,wecan 𝐿={0,1,...,𝑚} ℎ decomposition of the finite chain . show that ∩ 𝜌𝑖 =𝜀𝐿.ByLemma2,𝐿 is isomorphic to a 𝐿={0,1,...,𝑚} 𝐽 𝑖=1 Since for a finite chain ,theset of all the subdirect product of 𝐿𝑖 (𝑖=1,2,...,ℎ). join irreducible elements of 𝐿 is just equal to 𝐿, and also note that a finite chain must be a finite distributive lattice, then we Further, denote ℎ = ⌈𝑚/(𝑘−1)⌉,whichistheleastinteger immediately obtain the following corollary by Theorem 9. greater than or equal to 𝑚/(𝑘 ,andtake− 1) 𝐿𝑖 ={0,...,(𝑘− 1)𝑖−1,(𝑘−1)𝑖}(𝑖=1,2,...,ℎ−1)and 𝐿ℎ ={0,...,(𝑘−1)ℎ− Corollary 14. Let 𝐿 = {0,1,...,𝑚} be a finite chain and 1, (𝑘 − 1)ℎ} when ℎ = ⌈𝑚/(𝑘 − 1)⌉ = 𝑚/(𝑘; −1) 𝐿ℎ = {0, (𝑘 − 𝐿𝑖 ={0,𝑖}(𝑖=1,2,...,𝑚).Then𝐿 is isomorphic to a subdirect 1)(ℎ−1),...,𝑚}when ⌈𝑚/(𝑘−1)⌉>𝑚/(𝑘−1). Generally, product of subdirect irreducible elements 𝐿𝑖 (𝑖=1,2,...,𝑚). we will have the following theorem. Discrete Dynamics in Nature and Society 5

ℎ Theorem 17. Let 𝐿 = {0,1,...,𝑚} be a finite chain. Then Now, let 𝜌𝑖 = Ker𝜙𝑖. Next, we show that 𝜌=∩𝑖=1𝜌𝑖 =𝜀𝐿. 𝐿 is isomorphic to a subdirect product of 𝐿𝑖 (𝑖 = 1,2,...,ℎ) Assume that 𝑎=𝑝𝑙 , 𝑏=𝑝𝑘 ∈𝐿,and(𝑎, 𝑏);then ∈𝜌 𝑠𝑗 𝑠𝑡 constructed as above. we have (𝑎, 𝑏)𝑖 ∈𝜌 (𝑖 = 1,2,...,ℎ),clearly,(𝑎, 𝑏)𝑙 ∈𝜌 and (𝑎, 𝑏) ∈𝜌 (𝑎, 𝑏) ∈𝜌 𝑏≥𝑝 =𝑎 𝑘.By 𝑙,wehave 𝑙𝑠 .Similarly,by Proof. For 𝑖=1,2,...,ℎ, define 𝜙𝑖 :𝐿→𝐿𝑖 by 𝑗 (𝑎, 𝑏)𝑘 ∈𝜌 ,wehave𝑎≥𝑝𝑘 =𝑏.Hence,weobtain𝑎=𝑏. 𝑠𝑡 {(𝑘−1) 𝑖, 𝑎 ≥ (𝑘−1) 𝑖, Now, by Lemma 2, we immediately verify that 𝐿 is { { isomorphic to a subdirect product of 𝐿𝑖 (𝑖=1,2,...,ℎ). 𝑎𝜙 = 𝑎, (𝑘−1) 𝑖>𝑎≥(𝑘−1)(𝑖−1) , 𝑖 { (9) { {0, (𝑘−1)(𝑖−1) >𝑎≥0. 3. Conclusion

Then, it is a routine way to check that 𝜙𝑖 (𝑖= 1,2,...,ℎ)is a Subdirect decomposition of algebra is one of its quite general homomorphism. Now, let 𝜌𝑖 = Ker𝜙𝑖. Also, we can show that and important constructions. In this paper, we investigate ℎ ∩𝑖=1𝜌𝑖 =𝜀𝐿. By Lemma 2, we can complete our proof. some subdirect decompositions (including subdirect irreducible decompositions) of finite distributive lattices and Finally, we will give a more general subdirect decomposi- finite chains, and give a lot of concrete examples. Actually, tion of a finite chain. ℎ the main results in this paper are good complements of the Let 𝐿=⋃𝑖=𝑙 𝐿𝑖 be a finite chain with 𝑚+1elements where corresponding ones in [8]. 𝑚≥1.Here𝐿𝑖 ={0,𝑝𝑖 ,...,𝑝𝑖 } is a more than 2 elements 1 𝑠𝑖 subchain of 𝐿 satisfying 0=𝑝𝑖 <𝑝𝑖 < ⋅⋅⋅ < 𝑝𝑖 , 𝑠1 +𝑠2 + 0 1 𝑠𝑖 Conflicts of Interest ⋅⋅⋅+𝑠ℎ =𝑚,andalso𝐿𝑖 ⋂ 𝐿𝑗 ={0}if 𝑖=𝑗 ̸ . The authors declare that they have no conflicts of interest. ℎ Theorem 18. Let 𝐿=⋃𝑖=𝑙 𝐿𝑖 be a chain with 𝑚+1elements where 𝑚≥1and 𝐿𝑖 ={0,𝑝𝑖 ,...,𝑝𝑖 } are constructed as 1 𝑠𝑖 Acknowledgments above. Then 𝐿 is isomorphic to a subdirect product of 𝐿𝑖 (𝑖 = 1,2,...,ℎ). This work is supported by Grants of the NNSF of China (nos. 11501237, 11401246, 11426112, and 61402364); the Proof. For 𝑖=1,2,...,ℎ, define 𝜙𝑖 :𝐿→𝐿𝑖 by NSF of Guangdong Province (nos. 2014A030310087, 2014A030310119, and 2016A030310099); the Outstanding {𝑝𝑖 ,𝑎≥𝑝𝑖, Young Teacher Training Program in Guangdong Universities 𝑠𝑖 𝑎𝜙𝑖 = { (10) (no. YQ2015155); and Scientific Research Innovation Team 𝑝𝑖 ,𝑎≥𝑝𝑖 ,𝑎≥𝑝̸ 𝑖 (0 ≤ 𝑠𝑗 ≤𝑠𝑖 −1). { 𝑠𝑗 𝑠𝑗 𝑠𝑗+1 ProjectofHuizhouUniversity(hzuxl201523). 𝜙 (𝑖= 1,2,...,ℎ) Then, it is a routine way to verify that 𝑖 References is a homomorphism. 𝜙 (𝑖=1,2,...,ℎ) Firstly, 𝑖 is clearly a mapping. [1] J. S. Golan, Semirings and Their Applications, Kluwer Academic Secondly, for any 𝑎, 𝑏,wewillshowthat ∈𝐿 (𝑎 ∨ 𝑏)𝜙𝑖 = Publishers, Dordrecht, Netherlands, 1999. 𝑎𝜙 ∨𝑏𝜙 (𝑖=1,2,...,ℎ). 𝑖 𝑖 [2] Y. Chen, H. Wang, and H. Luo, “Decompositions of matrices over a finite chain,” International Journal of Pure and Applied (i) If 𝑎∨𝑏≥𝑝𝑖 ,then,since𝐿 is a finite chain, we have 𝑠𝑖 Mathematics,vol.75,no.2,pp.149–158,2012. 𝑎≥𝑝𝑖 or 𝑏≥𝑝𝑖 ,andthenweget(𝑎 ∨ 𝑏)𝜙𝑖 =𝑝𝑖 = 𝑠𝑖 𝑠𝑖 𝑠𝑖 [3] G. Birkhoff, Lattice Theory, XXV, American Mathematical 𝑎𝜙𝑖 ∨𝑏𝜙𝑖. Society Colloquium Publications, 1984. 𝑎∨𝑏≥𝑝 𝑎∨𝑏≥𝑝̸ (0 ≤ 𝑠 ≤𝑠−1) [4] S. Burris and H. P. Sankappanavar, A Course in Universal Alge- (ii) If 𝑖𝑠 , 𝑖𝑠 +1 𝑗 𝑖 ,then 𝑗 𝑗 bra (The Millennium Edition), 2000. 𝑎≥𝑝𝑖 or 𝑏≥𝑝𝑖 , 𝑎≥𝑝 ̸ 𝑖 and 𝑏≥𝑝 ̸ 𝑖 ,andthen 𝑠𝑗 𝑠𝑗 𝑠𝑗+1 𝑠𝑗+1 [5]T.S.Blyth,Lattices and Ordered Algebraic Structure,Springer- (𝑎 ∨ 𝑏)𝜙𝑖 =𝑝𝑖 =𝑎𝜙𝑖 ∨𝑏𝜙𝑖. 𝑠𝑗 Verlat London Limited, London, UK, 2005. [6] G. Birkhoff, “Subdirect unions in universal algebra,” Bulletin of (𝑎∧𝑏)𝜙 =𝑎𝜙∧𝑏𝜙 (𝑖=1,2,...,ℎ) Thirdly, we show that 𝑖 𝑖 𝑖 the American Mathematical Society,vol.50,pp.764–768,1944. 𝑎, 𝑏 ∈𝐿 for any . [7] H.-J. Bandelt and M. Petrich, “Subdirect products of rings and distributive lattices,” Proceedings of the Edinburgh Mathematical (i) If 𝑎∧𝑏≥𝑝𝑖 ,then𝑎≥𝑝𝑖 and 𝑏≥𝑝𝑖 ,andthenwe 𝑠𝑖 𝑠𝑖 𝑠𝑖 Society. Series II,vol.25,no.2,pp.155–171,1982. get (𝑎 ∧ 𝑏)𝜙𝑖 =𝑝𝑖 =𝑎𝜙𝑖 ∧𝑏𝜙𝑖. 𝑠𝑖 [8] Y. Chen, X. Zhao, and L. Yang, “On n × n matrices over a finite 𝑎∧𝑏≥𝑝 𝑎∧𝑏≥𝑝̸ (0 ≤ 𝑠 ≤𝑠−1) distributive lattice,” Linear and Multilinear Algebra,vol.60,no. (ii) If 𝑖𝑠 , 𝑖𝑠 +1 𝑗 𝑖 ,then 𝑗 𝑗 2,pp.131–147,2012. 𝑎≥𝑝𝑖 and 𝑏≥𝑝𝑖 , 𝑎≥𝑝 ̸ 𝑖 or 𝑏≥𝑝 ̸ 𝑖 ,andthen 𝑠𝑗 𝑠𝑗 𝑠𝑗+1 𝑠𝑗+1 (𝑎 ∧ 𝑏)𝜙𝑖 =𝑝𝑖 =𝑎𝜙𝑖 ∧𝑏𝜙𝑖. 𝑠𝑗 Summing up all the discussions above, we have shown that 𝜙𝑖 (𝑖=1,2,...,ℎ)is a homomorphism. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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