International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 157169, 8 pages

doi:10.1155/2012/157169

Research Article

Noetherian and Artinian Lattices

Derya Keskin Tütüncü,¹ Sultan Eylem Toksoy,² and Rachid Tribak³

¹Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, Turkey

²Department of Mathematics, İzmir Institute of Technology, 35430 Urla, İzmir, Turkey

³Centre Pédagogique Régional (CPR) Tanger, Avenue My Abdelaziz Souani, BP 3117, Tangier 90000, Morocco

Received 30 March 2012; Accepted 12 April 2012

Academic Editor: Palle E. Jorgensen

Copyright © 2012 Derya Keskin Tütüncü 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.

Abstract.

It is proved that if L is a complete modular lattice which is compactly generated, then Rad(L)/0 is Artinian if, and only if for every small element a of L, the sublattice a/0 is Artinian if, and only if L satisfies DCC on small elements.

1. Introduction

By a lattice we mean a partially ordered set such that every pair of elements , in has a greatest lower bound (or a meet) and a least upper bound (or a join) ; that is,

(i), , and for all with , , (ii), , and for all with , .

Note that, for given , , and are unique, and

Let (or just ) be any lattice. Given , , we set

Then is a partially ordered set; moreover, for any , , , have greatest lower bound and least upper bound . We call the opposite lattice of , and denote it by .

Let be any lattice. Let in . We define

(Sometimes frac is denoted by .)

A lattice   has a least element if there exists such that . In this case, is uniquely defined and is usually denoted by 0. The lattice has a greatest element if there exists such that . In this case, is uniquely defined and is usually denoted by 1. A lattice is called complete if every subset of has a meet and a join, and it is called modular if for all , , in with . For more information about lattice theory, refer to [1–3].

Throughout this paper will be a complete modular lattice. An element is called an essential element if for every nonzero element . An element is said to be small if is an essential element of the opposite lattice . Let denote the set of all essential elements of . The set of all small elements of will be denoted by .

A set is called a direct set if, for all , , there exists with . The lattice is said to be upper continuous if, for every direct set in and element , we have . On the other hand, is said to be lower continuous if for every inverse set (i.e., for all , in , there exists with and element , . We will call an element in finitely generated element (or compact element) if whenever , for some direct set in , then there exists such that . Note that 0 is always a finitely generated element of . It is known that an element is finitely generated if and only if for every nonempty subset of with there exists a finite subset of such that . A lattice is said to be finitely generated (or compact) if 1 is finitely generated. We call the lattice   compactly generated if each of its elements is a join of finitely generated elements (see [2]). Note that every compactly generated lattice is upper continuous (see, e.g., [4, Proposition 2.4]). Moreover, it is shown in [4, Exercises 2.7 and 2.9] that for every element of a compactly generated lattice , the sublattices and are again compactly generated. A lattice is called a finitely cogenerated (or cocompact) lattice, if for every subset of such that there is a finite subset of such that . An element is said to be finitely cogenerated (or cocompact) if the sublattice is a finitely cogenerated lattice. If and imply , then we say that is covered by (or   covers ). If 0 is covered by an element of , then is called an atom element of . A lattice is said to be semiatomic if 1 is a join of atoms in (see [4]). The meet of all maximal elements (different from 1) in is denoted by , and it is called the radical of (see [2]). If is compactly generated, then is the join of all small elements of (see [2, Theorem 8]). The join of all atoms of , denoted by , is called the socle of . The socle of a compactly generated lattice is equal to the meet of all essential elements (see [4, Theorem 5.1]).

A non-empty subset of is called an independent set if, for every and finite subset of with , . We say that a nonzero lattice has finite uniform (or Goldie) dimension if contains no infinite independent sets; equivalently, sup contains an independent subset of cardinality equal to . In this case is said to have uniform (or Goldie) dimension and this is denoted by . We shall say that has hollow (or dual Goldie) dimension , provided the opposite lattice has uniform dimension . The lattice is said to be Artinian (noetherian) if satisfies the descending (ascending) chain condition on its elements. A lattice will be called an - lattice if, for each , there exists such that and .

In Section 2 we mainly prove that a lattice is noetherian if and only if is -complemented and every essential element of is finitely generated (Corollary 2.4). In Section 3 we generalize Theorem  5 in [5] to lattice theory (Theorem 3.7).

2. Noetherian Lattices

The following lemma was given us by Patrick F. Smith from his unpublished notes.

Lemma 2.1. Let be a lattice. Consider the following statements.

(i) is noetherian.(ii) has finite uniform dimension.(iii) is -complemented.

Then .

Proof. Suppose is noetherian but that does not have finite uniform dimension. Then there exists an infinite independent set of nonzero elements . Consider the ascending chain in . Because is noetherian, there exists a positive integer such that . This implies that , a contradiction. Therefore has finite uniform dimension.

Let . If , we are done. If , then there exists such that . If , we are done. Otherwise, there exists such that . Repeating this argument we produce an independent set . Thus this process must stop, so there exists such that and .

Remark 2.2. Note that if is a finitely generated element of a lattice , then for every non-empty set with there exists a finite subset of such that .

Proposition 2.3. Let be a lattice such that is finitely generated for every . Then the following are equivalent.

(i) is noetherian.(ii) has finite uniform dimension.(iii) is -complemented.

Proof. We only need to prove by Lemma 2.1. Let be a nonzero element in . By (iii), there exists an element of such that and . By hypothesis, is finitely generated. Let for a nonempty set in . Then . Since is finitely generated, for a finite subset of . Since is modular, we have . Therefore every element in is finitely generated. Hence is noetherian by [4, Proposition 2.3].

Corollary 2.4. A lattice is noetherian if and only if is -complemented and every essential element of is finitely generated.

Lemma 2.5. Every upper continuous lattice is -complemented.

Proof. Let . Let . Clearly, . Let be a chain in and let . Then . By Zorn’s lemma, contains a maximal member . Then . Suppose that for some . Then , and hence . Since , we have and . Thus . It follows that . Therefore is -complemented.

Corollary 2.6. Let be an upper continuous lattice. Then is noetherian if and only if every essential element in is finitely generated.

Lemma 2.7 (see [4, Lemmas 7.3 and 7.5]). Let be a lattice and a positive integer. Then

(i)if , then for every ;(ii)if , then .

As an easy observation of Lemma 2.7, we can give the following two results.

Proposition 2.8 (see cf. [5, Proposition 2]). Let be a compactly generated lattice. Then is noetherian if and only if satisfies on small elements.

Proof. By [2, Theorem 8].

By assumption, contains a maximal small element . Since is small in , . Suppose that . Then there exists a small element of such that . On the other hand, is a small element of by Lemma 2.7(ii). By the maximality of , we have . This gives , a contradiction. Thus . By Lemma 2.7(i), . Consequently, is noetherian.

Proposition 2.9 (see cf. [5, Proposition 3]). Let be a compactly generated lattice. Then the following are equivalent.

(i) has finite uniform dimension.(ii)There exists a positive integer such that for every small element of we have .(iii) does not contain an infinite independent set of nonzero small elements.

Proof. Let be a small element of . By [2, Theorem 8], . Since , has finite uniform dimension. The rest is clear.

Let be an infinite independent set of nonzero small elements of . By Lemma 2.7(ii), , and , a contradiction.

Suppose that does not have finite uniform dimension. Then there exists an infinite independent set of nonzero elements of . Let . Since is compactly generated, there exists a nonzero finitely generated element of such that . So by Lemma 2.7, . Therefore is an infinite independent set of nonzero small elements of , a contradiction. Thus has finite uniform dimension.

3. Artinian Lattices

Lemma 3.1. Let be a compactly generated semiatomic lattice. Then the following are equivalent.

(i) is finitely generated.(ii) is finitely cogenerated.(iii)1 is a finite independent join of atoms.(iv) is Artinian.

Proof. By [4, Theorem 11.1].

By [4, Proposition 11.2].

Note that if is an atom in , then is Artinian. Assume that such that the join is independent and each is atom in . Since each is Artinian, is Artinian, and hence is Artinian.

Lemma 3.2. Let be a compactly generated lattice which satisfies on small elements. If is a finitely generated element of , then is Artinian.

Proof. Let be a finitely generated element of . Then implies that for some finite subset of . By Lemma 2.7, . By assumption and Lemma 2.7(i), is Artinian.

Lemma 3.3. Let be a compactly generated lattice which satisfies on small elements. Then, for every , is an essential element of .

Proof. Let , and let . Let such that . Assume that . Since is compactly generated, there exists a nonzero finitely generated element in such that but . By Lemma 3.2, is Artinian. Then implies that is a nonzero Artinian sublattice. By [4, Proposition 1.4], has an atom element . Note that . Since is atom in , we have . Thus . This contradicts the fact that . Therefore and . This completes the proof.

Lemma 3.4. Let be an element of a compactly generated lattice . If is a finitely generated element of , then is a finitely generated element of .

Proof. Since is compactly generated, where is a set of finitely generated elements in . Since is a finitely generated element of , for some elements of . Therefore is a finitely generated element of .

Lemma 3.5. Let be a compactly generated lattice which satisfies on small elements. Suppose that the set

is nonempty. Then:

(1)the set has a minimal member which is a small element of ;(2)if , then is not a finitely generated element of and is a small element of .

Proof. (1) Let be any chain in . Let . If , then is finitely cogenerated. Therefore for some , a contradiction. By Zorn’s Lemma, has a minimal member . Let . By Lemma 3.3, . Thus is not a finitely generated element of by [4, Theorem 11.2]. Let with . Then . It follows that . Suppose that . Then is finitely cogenerated. Let . Then is finitely generated in by [4, Theorem 11.2]. Therefore is Artinian by Lemma 3.1. Since is a sublattice of , we have . Thus . Since is Artinian, is also Artinian by [4, Proposition 1.5]. This implies that is Artinian, and hence is a finitely generated element of by Lemma 3.1. Since is compactly generated, is a finitely generated element of (see Lemma 3.4), a contradiction. So and hence . This gives .

(2) Note that is not a finitely generated element of as we prove in (1). Let such that . Note that is a semiatomic lattice. Then is also semiatomic by [4, Corollary 6.3]. Therefore,

This implies that is semiatomic. Suppose that . By [4, Lemma 6.12], there exists a maximal element of . Clearly, is a maximal element of and . Thus . But . Then , a contradiction. It follows that . Since , we have . Thus .

Remark 3.6. By dualizing [6, Theorem 3.4], we have the fact that if is upper continuous and is Artinian for every small element of , then is Artinian. Therefore for compactly generated lattices in Theorem 3.7 holds, but our aim is to give a proof in a different way. We should call attention to the fact that need not to be the radical of any upper continuous lattice .

Theorem 3.7 (see cf. [5, Theorem 5]). Let be a compactly generated lattice. Then the following are equivalent.

(i) is Artinian.(ii)For every small element of the sublattice is Artinian.(iii) satisfies on small elements.

Proof. Clear by [2, Theorem 8].

This is immediate.

Suppose that is not Artinian. By [4, Proposition 11.2], there exists an element in with such that is not finitely cogenerated. By Lemma 3.5, the set

has a minimal member such that and is not a finitely generated element of . By (iii) and Lemma 2.7(i), is Artinian. By Lemma 3.1, is finitely generated. Therefore is a finitely generated element of by Lemma 3.4. This is a contradiction. Therefore is Artinian.

Corollary 3.8. Let be a compactly generated lattice. If is finitely cogenerated for every small element of , then is Artinian.

Proof. Consider the descending chain

of small elements of . Put . Thus is small in . By assumption, is finitely cogenerated. So there exists an integer such that . Hence has DCC on small elements. By Theorem 3.7, is Artinian.

Let and be elements of . Then is called a supplement of in if is minimal with respect to . Equivalently, is a supplement of if and only if and (see [4, Proposition 12.1]). The lattice is said to be supplemented if every element of has a supplement in .

The following result may be proved in much the same way as [5, Lemma 6], and is a semiatomic lattice by [4, Proposition 12.3] already.

Lemma 3.9. Let be a compactly generated supplemented lattice with on supplement elements. Then is a finitely generated semiatomic lattice.

By using Theorem 3.7 and Lemma 3.9, we get the following theorem.

Theorem 3.10. Let be a compactly generated lattice. Then is Artinian if and only if is supplemented and satisfies DCC on supplement elements and small elements.

Proof. The necessity is clear. Conversely, suppose that is a supplemented lattice which satisfies DCC on supplement elements and small elements. By Theorem 3.7, is Artinian, and by Lemmas 3.1 and 3.9, is Artinian. Thus is Artinian.

Acknowledgments

The second author has been supported by the TÜBİTAK (The Scientific and Technological Research Council of Turkey) within the program numbered 2218. She would like to thank TÜBİTAK for their support. Also the authors would like to thank Professor Patrick F. Smith (Glasgow University) for his helpful comments on the paper.

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