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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 :→Π∈ is subdirect if is asubdirectproductof.Atthistime,wealsosaythat (,+,⋅) A semiring is an algebraic structure consisting of has subdirect decomposition of or is isomorphic to the a nonempty set together with two binary operations + and subdirect product of {}∈. ⋅ (, +) (, ⋅) on such that and are semigroups connected A semiring is called subdirectly irreducible if for every ( + ) = + ( + ) = by distributivity, that is, and subdirect embedding :→Π∈,thereisan∈such + , , ∈ ,forall [1, 2]. A semiring is called a partially that ∘ :→ 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 ≤ is a partial order on satisfying the following condition: subdirectly irreducible. , , , ∈ ≤ ≤ + ≤ + for any ,if and ,then and From [4–7] we know that the subdirect product is a quite ≤ . 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 = tributive laws [3]. In the following, we will denote on the famous Birkhoff representation theorem. Formulated ⟨,+,⋅,0,1⟩ 0 1 as a finite distributive lattice, where and are in terms of semirings, it asserts that every semiring can be theleastandthegreatestelementsof ,respectively,andthe represented as a subdirect product of subdirectly irreducible addition and the multiplication on are defined as follows: semirings, and it can often reduce studying the structure +=∨=max {,} , of semirings from a given class to studying subdirectly irreducible members of this class. Also, there is a third way ⋅=∧=min {,} (1) of approaching subdirect decompositions which is based on ∀, ∈ . another Birkhoff theorem verified in [4], which, in terms of semirings, says that a semiring isasubdirectproductofa Also, we denote = {0,1,...,}as a finite chain with usual family of semirings {}∈ if and only if there exists a family ordering [4]. Clearly, both the finite distributive lattices and of factor congruences {}∈ on such that ⋂∈ = and the finite chains are partially ordered semirings. / = for each ∈;here, is the identity congruence A semiring issaidtobeasubdirectproductofan on . indexed family ( ∈ ) ofsemiringsifitsatisfies≤Π∈ The main aim of this paper is to investigate the subdirect and = for each ∈. decompositions of a special class of semirings called finite 2 Discrete Dynamics in Nature and Society distributive lattices. Although some subdirect decomposi- and ≤⋁=1 ( ∈,1≤≤), then there exists 0, 1≤ ≤ ≤ . tions of a finite distributive lattice are discussed in [8], and 0 ,suchthat 0 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. 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 .If∈and ≤ [8]. For notations and terminologies occurred but not men- ⋁=1 ( ∈,1≤≤), then there exists 0, 1≤0 ≤, ≤ tioned in this paper, the readers are referred to [1, 4]. such that 0 . Now, we can give some subdirect irreducible decomposi- 2. Subdirect Decompositions of tions of a finite distributive lattice . Finite Distributive Lattices Theorem 9. Let be a finite distributive lattice with +1 To obtain our main results in this section, we will also need elements where ≥1and ={0,1,2,...,ℎ}(≥ℎ≥1) the following lemmas and concepts. are the set of all the join irreducible elements of .Then is isomorphic to a subdirect product of subdirect irreducible Lemma 1 (see [4]). In the equational class of distributive elements ={0,} (=1,2,...,ℎ). lattices the only nontrivial subdirectly irreducible algebra is the two-element chain. Proof. For =1,2,...,ℎ, define :→ by Let be a semiring and Con be the set of all congruences { ,≥, on .ByLemma8.2in[4],wehavethefollowinglemma. = { (4) 0, . { otherwise Lemma 2. If ∈ Con for ∈and ∩=1 =, then the natural homomorphism Then, it is a routine way to verify that (= 1,2,...,ℎ)is a homomorphism. :→Π∈ (2) Firstly, (=1,2,...,ℎ)is clearly a mapping. Secondly, for any , ,wewillshowthat ∈ ( ∨ ) = defined by ∨ (=1,2,...,ℎ). ()() = (i) If ∨≥,then,byCorollary8,≥ or ≥, (3) andthenweget( ∨ ) = = ∨. ∨≱ ≥ ̸ ≥ ̸ ( ∨ is a subdirect embedding. (ii) If ,then and ,andthen ) =0=0∨0= ∨. In the following, we will discuss the subdirect decompo- (∧) =∧ (=1,2,...,ℎ) sitions of a finite distributive lattice. Thirdly, we show that for any , . ∈ Definition 3 (see [4]). Let be a lattice. The element ∈is (i) If ∧≥,then≥ and ≥.Thus,wehave called join irreducible of if for , , ∈ =∨implies ( ∧ ) = = ∧ = ∧. =or =. (ii) If ∧ ≱ ,then≥ ̸ or ≥,andthen(∧) = 0= ∧. Example 4. Let 36 denote the divisible lattice which is generated by all the positive factor of 36; then the set of all Summing up all the discussions above, we have shown the join irreducible of 36 is {1,2,3,4,9}. that (=1,2,...,ℎ)is a homomorphism. = =∩ℎ = Definition 5 (see [4]). Let be a finite distributive lattice Now, let Ker . Next, we show that =1 . , ∈ (, ) ∈ and ∈. If there exist join irreducible elements ( = Assume that and ,thenwehave (, ) ∈ ( = 1,2,...,ℎ) 1,2,...,)such that =⋁=1,where ⩽¡ for any 1≤ ̸= . By Lemma 6, there exist the join irreducible decompositions of and . Assume that ≤,thenwecall⋁=1 the join irreducible decomposition = ⋁ ⋁⋅⋅⋅⋁ and = ⋁ ⋁⋅⋅⋅⋁ , of . 1 2 1 2 where , ,..., , , ,..., ∈.Thenweget≥ , ,...,1 2 ≥1 2 , ,..., (, ) ∈ Lemma 6 (see [4]). Let be a finite distributive lattice and 1 2 and 1 2 .Since ( = 1,2,...,ℎ) ≥, ,..., ≥ be the set of all the join irreducible elements of .Thenfor ,wehave 1 2 , ∈, , ,..., = ⋁ ⋁⋅⋅⋅⋁ ≤= any has a unique join irreducible decomposition and 1 2 ,andthen 1 2 , =⋁ ⋁ ⋁⋅⋅⋅⋁ ≤ = ∈,≤ . 1 2 .Hence,weobtain . Now, by Lemmas 1 and 2, we immediately verify that Lemma 7 (see [4]). Let be a finite distributive lattice and is isomorphic to a subdirect product of subdirect irreducible be the set of all the join irreducible elements of .If∈ elements (=1,2,...,ℎ). Discrete Dynamics in Nature and Society 3 1 1 c c b a ab 0 Figure 2 0 Figure 1 1 Example 10. Let ={0,,,,1}be a finite distributive lattice d as the Hasse diagram shown in Figure 1. Clearly, ={0,,,1}.Now,wecantake1 ={0,}, 2 = b c {0, },and3 = {0, 1} and define 3 a :→∏, =1 0 0 → (0, 0, 0) , Figure 3 → (, 0,) 0 , (5) → (0, ,) 0 , satisfying 0= < <⋅⋅⋅< , 1 +2 +⋅⋅⋅+ =ℎ,and → (, ,) 0 , 0 1 ⋂ ={0}if = ̸ .Notethatthemaximalchainfrom0toa 1 → (, ,) 1 . maximal element of may be not unique, and the expression ⋃= may be not unique in general. Then it is a routine way to check that is a subdirect 3 Further, we can take = ⋃= (1≤≤), where embedding homomorphism from to ∏=1.Hence, is ={0, ,..., } is a more than 2 elements subchain isomorphic to a subdirect product of subdirect irreducible 1 ( = 1, 2, 3) 0= < elements . of (and also a subchain of )andsatisfying 0 < ⋅⋅⋅ < , 1 +2 +⋅⋅⋅+ =,and ⋂ ={0} In general, if we replace the finite distributive lattice 1 = ̸ =⋃ =⋃ ⋃ with a finite lattice, we cannot get the corresponding if . Clearly, we have = = = ,where ={0, ,..., } is a subchain of and constructed as subdirect irreducible decomposition of . 1 above. Example 11. Let ={0,,,,1}be a finite lattice as the Hasse diagram shown in Figure 2. Example 12. Let be a finite distributive lattice whose Hasse = {0,,,} ={0,} Clearly, .Now,ifwetake 1 , diagram given as shown in Figure 3. Clearly, ={0,,,,1}, 2 ={0,},and3 ={0,}, then there is not any existing and we can take =1 ⋃ 2,where1 = {0, , , 1} and ∏3 subdirect embedding homomorphism from to =1 . 2 ={0,}(also, we can take 1 = {0, , , 1} and 2 ={0,}). Hence, is not isomorphic to a subdirect product of subdirect If we take 11 = {0, , ,} 12 = {0, 1},and21 ={0,}, (=1,2,3) irreducible elements . then =11 ⋃12 ⋃21.Ifwetake11 ={0,}, 12 = {0, }, 13 = {0, 1},and21 ={0,},thenwecanget= Next, we will discuss more general subdirect decomposi- 11 ⋃12 ⋃13 ⋃21. tions of a finite distributive lattice. +1 Let be a finite distributive lattice with elements Theorem 13. Let be a finite distributive lattice with +1 ≥1 ={0,, ,..., }(≥ ℎ ≥ 1) where and 1 2 ℎ are elements where ≥1and ={0,1,2,...,ℎ}(≥ℎ≥1) the set of all the join irreducible elements of .Then can be are the set of all the join irreducible elements of .Then is expressed as ⋃= ;here ={0, ,..., } is a more than (1≤≤,1≤≤) 1 isomorphic to a subdirect product of 2 elements maximal subchain of (and also a subchain of ) which is constructed as above.
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