The Properties of Maximal Filters in Multilattices

Jacobson’s radical of a lter F is the intersection of all maximal lters containing F. We present several properties of maximal lters in multilattices. As a consequence of Zorn’s lemma, we prove that each proper lter is contained in a maximal lter. When the lter lattice is distributive, we prove that each maximal lter is prime. Finally, we determine Jacobson’s radical of lters in multilattices.


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] identi ed three types of ideals and then showed that in multilattices, some concepts such as associativity and distributivity cannot be de ned 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 de nition of ideal (rep. lter) suitable for congruences. en, they proved that the set of all lters of a multilattice is a lattice with respect to inclusion. In [9], Awouafack et al. show that for some multilattices, the lter lattice is distributive.
is allowed them to de ne a notion of distributivity in multilattices which they used to establish the existence of prime lters.
is work is in the same register and aims at providing the main properties of maximal lters in multilattices. Some results are very close to the classical cases while others show that more caution is needed. e paper is structured as follows: Section 2 recalls de nitions and preliminary results necessary to understand the paper. Section 3 brings out the newly established results. Definition 2. [1] Let (M; ≤ ) be a poset. M is said to be an ordered multilattice if the following conditions hold for all x, y, a ∈ M. M 1 : x ≤ a and y ≤ a imply there exists z ∈ x ⊔ y such that z ≤ a. M 2 : a ≤ x and a ≤ y imply there exists z ∈ x ⊓ y such that a ≤ z. e two hyperoperations ⊔ : satisfy the following properties called axioms of multilattices [2,3]. e next theorem shows how to go back and forth from ordered multilattice to algebraic multilattice. Theorem 1. [9] e following assertions are satisfied: x ⊔ y � multisup(x, y) and x ⊓ y � multinf (x, y) for all x, y ∈ M.

Definition 4. [8]
(1) M is said to be a complete multilattice if every subset of M has at least one multisupremum and at least one multinfimum. (2) A ⊔ −full multilattice (resp. ⊓ -full multilattice) is a multilattice in which x ⊔ y ≠ ∅ (resp. x ⊓ y ≠ ∅) for all x, y ∈ M. 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 F of M is said to be a filter if it satisfies the following conditions: e ideal is the dual concept of filter. So, a nonempty subset I of M is said to be an ideal if it satisfies the following conditions: e set of all filters of M will be denoted by F(M).
When M is not ⊔ -full, it is necessary to add the empty set to F(M) (lifting) in order to obtain a lattice. at 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: Also, define the sequence (X n ) n∈N , recursively as follows: X 0 � X, X 1 � X * and ∀ n ≥ 1, X n+1 � (X n ⊓ X n ) * . en, the filter of M generated by X, denoted by 〈X〉 is given by: Many of our proofs will be based on the following lemma. Lemma 1. [9] Let x, y ∈ M en, the following assertions are satisfied:

Remark 1
(i) e inclusion of (2) of Lemma 1 is in general strict.
For instance, in the multilattice of Example 1, we have (ii) e map defined on the power set of M by X↦〈X〉 is an algebraic closure operator. is gives us the means to understand some properties of the filter lattice. (iii) However, the map 〈.〉: M ⟶ 2 M given by x↦〈x〉 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, 〈c 1 〉 � 〈c 2 〉 � M but c 1 ≠ c 2 . 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) K is said to be prime if for two filters (resp. ideals) A maximal filter is any maximal element of F(M)/ M { }.

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. e following result allows us to understand this difference. An illustration will be given in Example 1.

Theorem 3. Let M be a multilattice and let F be a filter of M.
en, the following assertions are satisfied: (1) If F is finite, then F is an upset.
(2) If F is finitely generated, then F is principal.
(3) If F is generated by a subset with least element, then F principal. (4) If F is not finitely generated, then F contains an infinite chain without the least element.
us, there exists i 2 ∈ F such that i 2 ≤ x 1 and i 2 ≤ x 2 and then, there exists i 3 ∈ F such that i 3 ≤ i 2 and i 3 ≤ x 3 , by inference, there exists i n ∈ F such that i n ≤ i n−1 and i n ≤ x n . We claim that i n ≤ x i for all i ∈ 1, 2, . . . , n { }. is implies F � ↑i n . For (2), let X � x 1 , x 2 , . . . , x n , n ∈ N * such that 〈X〉 � F. We form a sequence z i i�n i�1 as follows: en, X ⊆ ↑z n and z 1 , z 2 , . . . , z n ⊆ F. erefore, 〈z n 〉 � F and we obtain the desired conclusion.
For (4), let X � x i i∈N be an infinite subset without least element such that 〈X〉 ⊆ F. We form a sequence, z i i∈N as follows: z 0 � x 0 and for all i ≥ 1, z i ∈ x i ⊓ z i−1 . en, C � z i i∈N is an infinite chain of M contained in C. However, if C is finite it will has least element. us, F contains an infinite chain without least element.

□
We obtain the following result as a consequence of (4) of eorem 3.

□
In the multilattice of Example 1, ∪ i∈N ↑a i is a maximal filter which contains no element Δ.

Corollary 3. Let M be a coherent multilattice and let M be a maximal filter of M.
en, there exists α ∈ Δ such that M ⊆ ↑α. Example 1. Let us consider the multilattice M 1 which is schematized in Figure 1: We can easily verify that (i) F � ↑α, α ∈ ⊤, a i , b j , ⊥ is a principal filter of M 1 generated by α { }. (ii) F � ∪ k∈N ↑β k , β k � a i or β k � b j is a filter of M 1 which is not principal. It is generated by any unbounded subset of β k k∈N .
{ is a proper filter of M containing F}. Clearly, F ∈ H so H is not empty. Let C be a chain in H and let K � ∪ J∈C J. It is obvious that F ⊆ K. We claim that K is a proper filter of M. Let a, x, y ∈ K, if x, y ∈ K, then x ∈ J 1 and y ∈ J 2 for some J 1 , J 2 ∈ C, since C is a chain, either J 1 ⊆ J 2 or J 2 ⊆ J 1 , if, say J 1 ⊆ J 2 , then x, y ∈ J 2 and since J 2 is a filter, erefore, K is a proper filter of M containing F. By Zorn's Lemma, H has a maximal element, which is cleary a maximal filter of M. Notice that K(F) is not empty according to eorem 4. It is therefore a filter since any intersection of filters is a filter.
Proof. For simplicity, let us note Let M be a maximal filter of M containing F and let x ∈ M, x ∈ L. By Lemma 2, either x ∈ M or there exists is the intersection of all maximal filters of M. Corollary 6. Let F 1 , F 2 be two filters of M. en, the following condition holds: Figure 1: Multilattice M 1 and its filter lattice F(M 1 ).
Remark 2. Keeping the same notations as in Proposition 2, we can easily verify that J M (⊤) ⊆ ∩ α∈Δ ↑α whenever M is a coherent multilattice.
One of the fundamental properties of the Jacobson's radical is the following: Proof. Let x ∈ J M (F 1 ) ∩ J M (F 2 ), that is x ∈ J M (F 1 ) and x ∈ J M (F 2 ) and let y ∈ M such that 〈x〉 ∨ 〈y〉 � M. en, there exists z 1 ∈ F 1 and z 2 ∈ F 2 such that e reverse inclusion holds from Corollary 6 since F 1 ∩ F 2 ⊆ F 1 and □ e previous results show that when F(M) is a distributive lattice, the binary relation ≼ defined on M by a ≼ b if and only if J M (b) ⊆ J M (a) 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.

Conclusion and Perspectives
roughout 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. is 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
e authors declare that they have no conflicts of interest. Journal of Mathematics 5