Abstract
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
A semiring is an algebraic structure consisting of a nonempty set together with two binary operations + and on such that and are semigroups connected by distributivity, that is, and , for all [1, 2]. A semiring is called a partially ordered semiring if it admits a compatible ordering , that is, is a partial order on satisfying the following condition: for any , if and , then and . A partially ordered semiring is said to be a totally ordered semiring if the imposed partial order is a total order [1, 2].
A distributive lattice is a lattice which satisfies the distributive laws [3]. In the following, we will denote as a finite distributive lattice, where and are the least and the greatest elements of , respectively, and the addition and the multiplication on are defined as follows:Also, we denote as a finite chain with usual ordering [4]. Clearly, both the finite distributive lattices and the finite chains are partially ordered semirings.
A semiring is said to be a subdirect product of an indexed family of semirings if it satisfies and for each .
An embedding is subdirect if is a subdirect product of . At this time, we also say that has subdirect decomposition of or is isomorphic to the subdirect product of .
A semiring is called subdirectly irreducible if for every subdirect embedding , there is an such that is an isomorphism. From the above definition, it is easy to see that any two-element semiring is subdirectly irreducible.
From [4–7] we know that the subdirect product is a quite general construction. As for as semirings concerned, there are several ways of approaching subdirect decompositions of semirings. In most cases they can be obtained from various semirings theoretical constructions. Another way is based on the famous Birkhoff representation theorem. Formulated in terms of semirings, it asserts that every semiring can be represented as a subdirect product of subdirectly irreducible semirings, and it can often reduce studying the structure of semirings from a given class to studying subdirectly irreducible members of this class. Also, there is a third way of approaching subdirect decompositions which is based on another Birkhoff theorem verified in [4], which, in terms of semirings, says that a semiring is a subdirect product of a family of semirings if and only if there exists a family of factor congruences on such that and for each ; here, is the identity congruence on .
The main aim of this paper is to investigate the subdirect decompositions of a special class of semirings called finite distributive lattices. Although some subdirect decompositions of a finite distributive lattice are discussed in [8], and the subdirect decompositions of a finite chain are studied in [2], the results in this paper will be more general. We will investigate some subdirect decompositions (including subdirect irreducible decompositions) of finite distributive lattices and finite chains, whose proofs are also different from [8].
For notations and terminologies occurred but not mentioned in this paper, the readers are referred to [1, 4].
2. Subdirect Decompositions of Finite Distributive Lattices
To obtain our main results in this section, we will also need the following lemmas and concepts.
Lemma 1 (see [4]). In the equational class of distributive lattices the only nontrivial subdirectly irreducible algebra is the two-element chain.
Let be a semiring and be the set of all congruences on . By Lemma 8.2 in [4], we have the following lemma.
Lemma 2. If for and , then the natural homomorphism defined by is a subdirect embedding.
In the following, we will discuss the subdirect decompositions of a finite distributive lattice.
Definition 3 (see [4]). Let be a lattice. The element is called join irreducible of if for , implies or .
Example 4. Let denote the divisible lattice which is generated by all the positive factor of 36; then the set of all the join irreducible of is
Definition 5 (see [4]). Let be a finite distributive lattice and . If there exist join irreducible elements such that , where for any , then we call the join irreducible decomposition of .
Lemma 6 (see [4]). Let be a finite distributive lattice and be the set of all the join irreducible elements of . Then for any has a unique join irreducible decomposition and .
Lemma 7 (see [4]). Let be a finite distributive lattice and be the set of all the join irreducible elements of . If and , then there exists , , such that
By Lemmas 6 and 7, we immediately obtain the following corollary.
Corollary 8. Let be a finite distributive lattice and be the set of all the join irreducible elements of . If and , then there exists , , such that .
Now, we can give some subdirect irreducible decompositions of a finite distributive lattice .
Theorem 9. Let be a finite distributive lattice with elements where and are the set of all the join irreducible elements of . Then is isomorphic to a subdirect product of subdirect irreducible elements .
Proof. For , define byThen, it is a routine way to verify that is a homomorphism.
Firstly, is clearly a mapping.
Secondly, for any , we will show that (i)If , then, by Corollary 8, or , and then we get (ii)If , then and , and then Thirdly, we show that for any .(i)If , then and . Thus, we have .(ii)If , then or , and then Summing up all the discussions above, we have shown that is a homomorphism.
Now, let . Next, we show that .
Assume that and , then we have . By Lemma 6, there exist the join irreducible decompositions of and . Assume that and , where . Then we get and . Since , we have , , and then , . Hence, we obtain .
Now, by Lemmas 1 and 2, we immediately verify that is isomorphic to a subdirect product of subdirect irreducible elements .
Example 10. Let be a finite distributive lattice as the Hasse diagram shown in Figure 1.
Clearly, . Now, we can take , , and and define Then it is a routine way to check that is a subdirect embedding homomorphism from to . Hence, is isomorphic to a subdirect product of subdirect irreducible elements .
In general, if we replace the finite distributive lattice with a finite lattice, we cannot get the corresponding subdirect irreducible decomposition of .
Example 11. Let be a finite lattice as the Hasse diagram shown in Figure 2.
Clearly, . Now, if we take , , and , then there is not any existing subdirect embedding homomorphism from to . Hence, is not isomorphic to a subdirect product of subdirect irreducible elements .
Next, we will discuss more general subdirect decompositions of a finite distributive lattice.
Let be a finite distributive lattice with elements where and are the set of all the join irreducible elements of . Then can be expressed as ; here is a more than 2 elements maximal subchain of (and also a subchain of ) satisfying , , and if . Note that the maximal chain from 0 to a maximal element of may be not unique, and the expression may be not unique in general.
Further, we can take (), where is a more than 2 elements subchain of (and also a subchain of ) and satisfying , , and if . Clearly, we have , where is a subchain of and constructed as above.
Example 12. Let be a finite distributive lattice whose Hasse diagram given as shown in Figure 3. Clearly, , and we can take , where and (also, we can take and ). If we take , , and , then . If we take , , , and , then we can get .
Theorem 13. Let be a finite distributive lattice with elements where and are the set of all the join irreducible elements of . Then is isomorphic to a subdirect product of which is constructed as above.
Proof. For , , define byThen, it is a routine way to verify that is a homomorphism.
Firstly, is clearly a mapping.
Secondly, for any , we will show that (i)If , then, by Corollary 8, or , and then we get (ii)If , then, by Corollary 8, or , and also we have and . Thus, Thirdly, we show that for any .(i)If , then and . Thus, we have (ii)If , , then we have and , and also or . Hence, Summing up all the discussions above, we have shown that is a homomorphism.
Now, let . Next, we show that .
Assume that and ; then we have . By Lemma 6, there exist the join irreducible decompositions of and . Assume that and , where . Then we get and . Since , we have and and then and . Hence, we obtain .
Now, by Lemma 2, we immediately verify that is isomorphic to a subdirect product of .
In the following, we will discuss the subdirect product decomposition of the finite chain .
Since for a finite chain , the set of all the join irreducible elements of is just equal to , and also note that a finite chain must be a finite distributive lattice, then we immediately obtain the following corollary by Theorem 9.
Corollary 14. Let be a finite chain and . Then is isomorphic to a subdirect product of subdirect irreducible elements .
In general, the subdirect decomposition manners of a finite chain into the subdirect irreducible elements can be various.
Example 15. Let be a finite chain. By the above corollary, we have shown that is isomorphic to a subdirect product of subdirect irreducible elements , where . On the other hand, we can also take and define then it is not hard to check that is also a subdirect embedding homomorphism from to . And so is isomorphic to a subdirect product of subdirect irreducible elements . Obviously, the two manners of subdirect decomposition of into the subdirect irreducible elements are different.
Next, we will denote , which is the least integer greater than or equal to , and take and , when , when . Also, we have the following theorem.
Theorem 16. Let be a finite chain. Then is isomorphic to a subdirect product of constructed above.
Proof. For , define byThen, we can check that is a homomorphism by a routine way. Now, let . Also, we can show that . By Lemma 2, is isomorphic to a subdirect product of .
Further, denote , which is the least integer greater than or equal to , and take and when ; when . Generally, we will have the following theorem.
Theorem 17. Let be a finite chain. Then is isomorphic to a subdirect product of constructed as above.
Proof. For , define byThen, it is a routine way to check that is a homomorphism. Now, let . Also, we can show that . By Lemma 2, we can complete our proof.
Finally, we will give a more general subdirect decomposition of a finite chain.
Let be a finite chain with elements where . Here is a more than 2 elements subchain of satisfying , , and also if .
Theorem 18. Let be a chain with elements where and are constructed as above. Then is isomorphic to a subdirect product of .
Proof. For , define byThen, it is a routine way to verify that is a homomorphism.
Firstly, is clearly a mapping.
Secondly, for any , we will show that (i)If , then, since is a finite chain, we have or , and then we get (ii)If , , then or , and , and then Thirdly, we show that for any .(i)If , then and , and then we get (ii)If , , then and , or , and then Summing up all the discussions above, we have shown that is a homomorphism.
Now, let . Next, we show that .
Assume that , , and ; then we have , clearly, and . By , we have . Similarly, by , we have . Hence, we obtain .
Now, by Lemma 2, we immediately verify that is isomorphic to a subdirect product of .
3. Conclusion
Subdirect decomposition of algebra is one of its quite general and important constructions. In this paper, we investigate some subdirect decompositions (including subdirect irreducible decompositions) of finite distributive lattices and finite chains, and give a lot of concrete examples. Actually, the main results in this paper are good complements of the corresponding ones in [8].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by Grants of the NNSF of China (nos. 11501237, 11401246, 11426112, and 61402364); the NSF of Guangdong Province (nos. 2014A030310087, 2014A030310119, and 2016A030310099); the Outstanding Young Teacher Training Program in Guangdong Universities (no. YQ2015155); and Scientific Research Innovation Team Project of Huizhou University (hzuxl201523).