Abstract
Jacobson’s radical of a filter F is the intersection of all maximal filters containing F. We present several properties of maximal filters in multilattices. As a consequence of Zorn’s lemma, we prove that each proper filter is contained in a maximal filter. When the filter lattice is distributive, we prove that each maximal filter is prime. Finally, we determine Jacobson’s radical of filters in multilattices.
1. Introduction
We owe the multilattice theory to Benado who, in his work on multistructures, had the intuition to generalize lattices by replacing the existence of unique lower (resp. upper) bound by the existence of maximal lower (resp. minimal upper) bounds [1]. Several authors will contribute to the consolidation of this theory by enriching it with the study of mathematical concepts. It is in this perspective that Klaućová [2] and Hansen [3] propose many characterizations of multilattices. In the same vein, Johnston [4] identified three types of ideals and then showed that in multilattices, some concepts such as associativity and distributivity cannot be defined in a canonical way. Indeed, the notion associativity is replaced by a less natural notion of m-associativity and the notion of distributivity remains nontrivial.
Following the example of Medina et al. [5, 6], several authors will be interested in this theory because of its numerous applications in information systems with uncertainty. Cabrera et al. [7] proposed an algebraization of multilattices and advocated in [8], a new definition of ideal (rep. filter) suitable for congruences. Then, they proved that the set of all filters of a multilattice is a lattice with respect to inclusion. In [9], Awouafack et al. show that for some multilattices, the filter lattice is distributive. This allowed them to define a notion of distributivity in multilattices which they used to establish the existence of prime filters. This work is in the same register and aims at providing the main properties of maximal filters in multilattices. Some results are very close to the classical cases while others show that more caution is needed.
The paper is structured as follows: Section 2 recalls definitions and preliminary results necessary to understand the paper. Section 3 brings out the newly established results.
2. Preliminaries
Let be a poset and let be a nonempty subset of . An element is said to be an upper bound of if it satisfies for all . Dually, is a lower bound of if it satisfies for all .
Definition 1. [1] A multisupremum (resp. multinfimum) of is a minimal upper (resp. maximal lower) bound of .
The set of multisuprema (resp. multinfima) of will be denoted by (resp. ). For all nonempty subsets of , and denote the following subsets:
Definition 2. [1] Let be a poset. is said to be an ordered multilattice if the following conditions hold for all . M: and imply there exists such that . M: and imply there exists such that .
The two hyperoperations and satisfy the following properties called axioms of multilattices [2, 3]. AM1: For all , ; AM2: For all , ; AM3: For all , ; AM4: For all , ; AM5: For all , .
Definition 3. [8] An algebraic multilattice is any triple which satisfies the properties AM1–AM5.
The next theorem shows how to go back and forth from ordered multilattice to algebraic multilattice.
Theorem 1. [9] The following assertions are satisfied:(i)If is an ordered multilattice, set Then, is an algebraic multilattice.(ii)Conversely if is a multilattice, set: Then, is an ordered multilattice.
We will simply write instead of or assuming that the underlined order and the corresponding hyperoperations are understood.
Definition 4. [8](1) is said to be a complete multilattice if every subset of has at least one multisupremum and at least one multinfimum.(2)A full multilattice (resp. -full multilattice) is a multilattice in which (resp. ) for all . A multilattice is said to be full if it is both full and full.(3)A coherent multilattice is a multilattice in which every chain is bounded.
In multilattices, there are several proposals for the definition of the ideal [4, 8]. In this paper, we use the one proposed in 2014 by Cabrera et al. [8] which is the only one adapted to the study of several mathematical concepts such as congruences and homomorphisms.
Definition 5. [8] A nonempty subset of is said to be a filter if it satisfies the following conditions: (F1): For all , ; (F2): For all and , ; (F3): For all such that , .The ideal is the dual concept of filter. So, a nonempty subset of is said to be an ideal if it satisfies the following conditions: (I-1): For all , ; (I-2): For all and , ; (I-3): For all such that , .
The set of all filters of will be denoted by .
Proposition 1. [8] Let be an -full multilattice. Then, is a lattice under the set inclusion.
The lattice is given by: and is the smallest filter of containing both and . When is not -full, it is necessary to add the empty set to (lifting) in order to obtain a lattice. That is is always a lattice ordered by set inclusion [9].
In [9], the authors described the filter generated by a nonempty subset by the substar operator as follows:
Theorem 2. [9] Let a nonempty subset of . Set
Also, define the sequence , recursively as follows: , and , . Then, the filter of generated by , denoted by is given by:
Many of our proofs will be based on the following lemma.
Lemma 1. [9] Let Then, the following assertions are satisfied:(1);(2);(3).
Remark 1. (i)The inclusion of (2) of Lemma 1 is in general strict. For instance, in the multilattice of Example 1, we have but .(ii)The map defined on the power set of by is an algebraic closure operator. This gives us the means to understand some properties of the filter lattice.(iii)However, the map given by is not an embedding, it will be an embedding if we are in front of a lattice. We can see this through Example 1. For instance, but . So, according to Cabrera et al. [8], in multilattices there are more elements than filters.
Before introducing our results, let us recall some commonly used concepts.
A filter (resp. an ideal) is said to be prime if for two filters (resp. ideals) , if then either or .
A maximal filter is any maximal element of .
3. Main Results
In a lattice, a filter is principal if and only if it is an upset whereas in multilattice the two notions appear to be distinct. The following result allows us to understand this difference. An illustration will be given in Example 1.
Theorem 3. Let be a multilattice and let be a filter of . Then, the following assertions are satisfied:(1)If is finite, then is an upset.(2)If is finitely generated, then is principal.(3)If is generated by a subset with least element, then principal.(4)If is not finitely generated, then contains an infinite chain without the least element.
Proof. For (1), let . Since , then, for all , there is such that and . Thus, there exists such that and and then, there exists such that and , by inference, there exists such that and . We claim that for all . This implies .
For (2), let , such that . We form a sequence as follows: and , . Then, and . Therefore, and we obtain the desired conclusion.
For (3), let be a subset of with a least element . Then, and it follows that . Since , we also have . This implies .
For (4), let be an infinite subset without least element such that . We form a sequence, as follows: and for all , . Then, is an infinite chain of contained in . However, if is finite it will has least element. Thus, contains an infinite chain without least element.
We obtain the following result as a consequence of (4) of Theorem 3.
Corollary 1. Any filter of a coherent multilattice is principal.
Let be multilattice with bottom . Let denotes the set of minimal elements of .
Proposition 2. Let be a maximal filter of . Then, contains at most one element of .
Proof. Let such that . Since is a filter we have . But necessarily, since and are minimal in . This implies , a contradiction.
In the multilattice of Example 1, is a maximal filter which contains no element .
Corollary 2. Let be a filter of such that . Then, is a maximal filter of verifying .
Corollary 3. Let be a coherent multilattice and let be a maximal filter of . Then, there exists such that .
Example 1. Let us consider the multilattice which is schematized in Figure 1:We can easily verify that(i), is a principal filter of generated by .(ii), or is a filter of which is not principal. It is generated by any unbounded subset of .
We notice that , which means that in multilattices, we generally have .
Lemma 2. Let be a coherent multilattice and let be a proper filter of . Then, is a maximal filter if and only if for all , there exists such that .
Proof. Suppose that is maximal. Since is coherent, there exists such that . It follows that for all due to the maximality of . From Lemma 1, we have .
Suppose that for all , for some .
Let be a proper filter of such that . Suppose that and let . Then, there exists such that . However, implies , so . This contradicts the fact that is a proper filter of . Hence, and it follows that .
Proposition 3. If is distributive, then every maximal filter of is prime.
Proof. For (1), let be a maximal filter of and let , be two filters of such that and neither nor . Then, due to the maximality of . It follows that . That is , which implies , a contradiction. Therefore, is a prime filter of .
The previous result is no longer true if is not distributive. For instance, for the multilattice of Figure 2, is not distributive since it contains a copy of the pentagon. We have but neither nor .
Theorem 4. Every proper filter of is contained in a maximal filter.
Proof. Let is a proper filter of containing . Clearly, so is not empty. Let be a chain in and let . It is obvious that . We claim that is a proper filter of . Let , if , then and for some , since is a chain, either or , if, say , then and since is a filter, ; if , then for some , thus ; if , then for some and then . Therefore, is a proper filter of containing . By Zorn’s Lemma, has a maximal element, which is cleary a maximal filter of .
Corollary 4. Let be a proper filter of . If such that for all . Then, there exists a maximal filter of containing and .
In [10], Coquand et al. defined Jacobson’s radical of a proper ideal of a distributive lattice as the intersection of maximal ideals of containing . The duality filter/ideal allows us to use this definition for filters. After all, a filter of a multilattice is nothing else than an ideal of the dual of .
Definition 6. For every proper filter of , let is a maximal filter of containing . Jacobson’s radical of is the filter of denoted and defined as follows: . It is the intersection of all maximal filters of containing .
Notice that is not empty according to Theorem 4. It is therefore a filter since any intersection of filters is a filter.
Theorem 5. Let be a proper filter of . Then,
Proof. For simplicity, let us noteLet be a maximal filter of containing and let , . By Lemma 2, either or there exists such that . If , then . Since , there exists such that . Since and is a filter of , that is . This contradicts the fact that is proper. Thus necessarily, and it follows that .
Suppose that and let . Then, there exists , such that and for all . Thus, there exists a maximal filter containing and . Hence, , and then that is , which contradicts the fact that is a proper filter of . This implies . Therefore, and the proof is completed.
Theorem 5 induces the following two results:
Corollary 5. If , we write instead of and we have
The intersection of all maximal filters of containing .
In particular, if , then,is the intersection of all maximal filters of .
Corollary 6. Let , be two filters of . Then, the following condition holds:
Remark 2. Keeping the same notations as in Proposition 2, we can easily verify that whenever is a coherent multilattice.
Remark 3. If has a bottom , then from Lemma 1, the set we use in Theorem 5 can alternatively be defined as follows:
Example 2. The multilattice whose Hasse diagram is given by Figure 2 has 13 filters of which 3 are maximal. Jacobson’s radicals of these filters are listed below:(i)For , ;(ii)For , ;(iii)For , ;(iv)For , .One of the fundamental properties of the Jacobson’s radical is the following:
Proposition 4. If is coherent and is distributive, then,
Proof. Let , that is and and let such that . Then, there exists and such that . Since is coherent, there exists such that . Therefore, since is distributive. This implies , and we obtain the inclusion . The reverse inclusion holds from Corollary 6 since and .
The previous results show that when is a distributive lattice, the binary relation defined on by if and only if is a preorder. We can look for the properties of this preorder in order to define on the multilattice a quotient which is neither a quotient by a filter nor a quotient by an ideal. Such an investigation could lead to the Heitmann dimension of multilattice.
4. Conclusion and Perspectives
Throughout this paper, we have studied the properties of maximal filters in multilattices. In addition to the classical properties related among others to the distributivity, we have introduced Jacobson’s radical of a filter. This gives tracks to define a particular quotient of multilattice and opens to the study of the Heitmann dimension of multilattice which we leave in perspective. With the ambition to study the representation of multilattices, especially Priestley’s representation, the study of the properties of maximal filters has been an important part of our project.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.