# The Properties of Maximal Filters in Multilattices

Authors: Daquin Cédric Awouafack; Emmanuel Fouotsa
Journal: Journal of Mathematics
 (2022)

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## Abstract

Jacobson’s radical of a filterF 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.


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## Body

## 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: Section2 recalls definitions and preliminary results necessary to understand the paper. Section 3 brings out the newly established results.


## 2. Preliminaries

LetM,≤ be a poset and let X⊆M be a nonempty subset of M. An element u∈M is said to be an upper bound of X if it satisfies x≤u for all x∈X. Dually, l∈M is a lower bound of X if it satisfies l≤x for all x∈X.Definition 1.
[1] A multisupremum (resp. multinfimum) of X is a minimal upper (resp. maximal lower) bound of X.The set of multisuprema (resp. multinfima) ofx,y will be denoted by x⊔y (resp. x⊓y). For all nonempty subsets X,Y of M, X⊔Y and X⊓Y denote the following subsets:(1)X⊔Y=∪x∈X,y∈Yx⊔yandX⊓Y=∪x∈X,y∈Yx⊓y.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.M1: x≤a and y≤a imply there exists z∈x⊔y such that z≤a.M2: a≤x and a≤y imply there exists z∈x⊓y such that a≤z.The two hyperoperations⊔:M2⟶PMa,b↦a⊔b and ⊓:M2⟶PMa,b↦a⊓b satisfy the following properties called axioms of multilattices [2, 3].AM1: For allx∈M, x⊔x=x,x⊓x=x;AM2: For allx,y∈M, x⊔y=y⊔x,x⊓y=y⊓x;AM3: For allx,y,z∈M, x≤y⇒x⊔y⊔z⊆x⊔y⊔z,x⊓y⊓z⊆x⊓y⊓z;AM4: For allx,y∈M, x⊔x⊓y=x⊓x⊔y=x;AM5: For allx,y∈M, x≤y⇔x⊔y=y⇔x⊓y=x.Definition 3.
[8] An algebraic multilattice is any triple M;⊔,⊓ 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)
IfM;≤ is an ordered multilattice, set(2)x⊔y=multisupx,yandx⊓y=multinfx,yfor allx,y∈M.Then,M;⊔,⊓ is an algebraic multilattice.(ii)
Conversely ifM;⊔,⊓ is a multilattice, set:(3)x≤y⇔x⊓y=x⇔x⊔y=y.Then,M;≤ is an ordered multilattice.We will simply writeM instead of M;≤,M;⊔,⊓ or M;≤,⊔,⊓ assuming that the underlined order and the corresponding hyperoperations are understood.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:(F1): For allx,y∈F, x⊓y⊆F;(F2): For alla∈M and x∈F, a⊔x⊆F;(F3): For allx,y∈M such that x⊔y∩F≠∅, x⊔y⊆F.
The ideal is the dual concept of filter. So, a nonempty subsetI of M is said to be an ideal if it satisfies the following conditions:(I-1): For allx,y∈I, x⊔y⊆I;(I-2): For alla∈M and x∈I, a⊓x⊆I;(I-3): For allx,y∈M such that x⊓y∩I≠∅, x⊓y⊆I.The set of all filters ofM will be denoted by ℱM.Proposition 1.
[8] Let M be an ⊔-full multilattice. Then, ℱM is a lattice under the set inclusion.The latticeℱM,∨,∧ is given by: F1∧F2=F1∩F2 and F1∨F2 is the smallest filter of M containing both F1 and F2. When M is not ⊔-full, it is necessary to add the empty set to ℱM (lifting) in order to obtain a lattice. That is FM=ℱM∪∅ 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 X a nonempty subset of M. Set(4)X∗≔∪a⊔b|a⊔b∩↑X≠∅,a,b∈M.Also, define the sequenceXnn∈ℕ, recursively as follows: X0=X, X1=X∗ and ∀n≥1, Xn+1=Xn⊓Xn∗. Then, the filter of M generated by X, denoted by X is given by: X=∪n∈ℕXnMany of our proofs will be based on the following lemma.Lemma 1.
[9] Let x,y∈M Then, the following assertions are satisfied:(1)
z∈x⊓y⇒z=x∨y;(2)
z∈x⊔y⇒z⊆x∧y;(3)
z,z′∈x⊔y⇒z=z′.Remark 1.
(i)
The inclusion of (2) of Lemma1 is in general strict. For instance, in the multilattice of Example 1, we have c1⊔c2=↑a2 but c1∧c2=M1.(ii)
The map defined on the power set ofM by X↦X is an algebraic closure operator. This gives us the means to understand some properties of the filter lattice.(iii)
However, the map.:M⟶2M 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, c1=c2=M but c1≠c2. 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) J,J′, if J∩J′⊆K then either J⊆K or J′⊆K.A maximal filter is any maximal element ofℱM/M.


## 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 Example1.Theorem 3.
LetM be a multilattice and let F be a filter of M. Then, the following assertions are satisfied:(1)
IfF is finite, then F is an upset.(2)
IfF is finitely generated, then F is principal.(3)
IfF is generated by a subset with least element, then F principal.(4)
IfF is not finitely generated, then F contains an infinite chain without the least element.Proof.
For (1), letF=x1,x2,…,xn. Since F⊓F⊆F, then, for all xi,xj∈F, there is xk∈F such that xk≤xi and xk≤xj. Thus, there exists i2∈F such that i2≤x1 and i2≤x2 and then, there exists i3∈F such that i3≤i2 and i3≤x3, by inference, there exists in∈F such that in≤in−1 and in≤xn. We claim that in≤xi for all i∈1,2,…,n. This implies F=↑in.
For (2), letX=x1,x2,…,xn, n∈ℕ∗ such that X=F. We form a sequence zii=1i=n as follows: z1=x1 and zi+1∈zi⊓xi+1, i=1,…,n−1. Then, X⊆↑zn and z1,z2,…,zn⊆F. Therefore, zn=F and we obtain the desired conclusion.
For (3), letX be a subset of M with a least element α. Then, X⊆↑α and it follows that X⊆α. Since α∈X, we also have α⊆X. This implies F=α.
For (4), letX=xii∈ℕ be an infinite subset without least element such that X⊆F. We form a sequence, zii∈ℕ as follows: z0=x0 and for all i≥1, zi∈xi⊓zi−1. Then, C=zii∈ℕ is an infinite chain of M contained in C. However, if C is finite it will has least element. Thus, F contains an infinite chain without least element.We obtain the following result as a consequence of (4) of Theorem3.Corollary 1.
Any filter of a coherent multilattice is principal.LetM be multilattice with bottom ⊥. Let Δ denotes the set of minimal elements of M/⊥.Proposition 2.
LetM be a maximal filter of M. Then, M contains at most one element of Δ.Proof.
Letα,β∈Δ such that α,β∈M. Since M is a filter we have α⊓β⊆M. But necessarily, α⊓β=⊥ since α and β are minimal in M/⊥. This implies M=M, a contradiction.In the multilattice of Example1, ∪i∈ℕ↑ai is a maximal filter which contains no element Δ.Corollary 2.
LetF be a filter of M such that F∩Δ=α. Then, F is a maximal filter of M verifying F=↑α.Corollary 3.
LetM be a coherent multilattice and let M be a maximal filter of M. Then, there exists α∈Δ such that M⊆↑α.Example 1.
Let us consider the multilatticeM1 which is schematized in Figure 1:(5)M1=ai,i∈ℕ∪bj,j∈ℕ∪ck,k=1,2,3,4∪⊥,⊤.
We can easily verify that(i)
F=↑α, α∈⊤,ai,bj,⊥ is a principal filter of M1 generated by α.(ii)
F=∪k∈ℕ↑βk, βk=ai or βk=bj is a filter of M1 which is not principal. It is generated by any unbounded subset of βkk∈ℕ.Figure 1
MultilatticeM1 and its filter lattice ℱM1.We notice thatM1=c1≠↑c1, which means that in multilattices, we generally have x≠↑x.Lemma 2.
LetM be a coherent multilattice and let M be a proper filter of M. Then, M is a maximal filter if and only if for all x∈M/M, there exists m∈M such that x⊓m=M.Proof.
⇒ Suppose that M is maximal. Since M is coherent, there exists y∈M such that M=y. It follows that x∨y=M for all x∈M/M due to the maximality of M. From Lemma 1, we have x⊓y=x∨y=M.
⇐ Suppose that for all x∈M/M, x∨m=M for some m∈M.
LetF be a proper filter of M such that M⊆F⊊M. Suppose that F/M≠∅ and let x∈F/M⊆M/M. Then, there exists m∈M such that x∨m=M. However, x,m∈F implies x∨m⊆F, so M=F. This contradicts the fact that F is a proper filter of M. Hence, F/M=∅ and it follows that F=M.Proposition 3.
IfℱM is distributive, then every maximal filter of M is prime.Proof.
For (1), letM be a maximal filter of M and let F, F′ be two filters of M such that F∧F′⊆M and neither F⊈M nor F′⊈M. Then, F∨M=F′∨M=M due to the maximality of M. It follows that F∨M∧F′∨M=M. That is F∧F′∨M=M, which implies M=M, a contradiction. Therefore, M is a prime filter of M.The previous result is no longer true ifℱM is not distributive. For instance, for the multilattice of Figure 2, ℱM2 is not distributive since it contains a copy of the pentagon. We have ↑a4∧↑b2=↑b4⊆↑c4 but neither ↑a4⊆↑c4 nor ↑b2⊆↑c4.Figure 2
The multilatticeM2 and its filter lattice ℱM2.Theorem 4.
Every proper filter ofM is contained in a maximal filter.Proof.
Letℋ=F⊆J⊊M|J 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∈CJ. 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∈J1 and y∈J2 for some J1,J2∈C, since C is a chain, either J1⊆J2 or J2⊆J1, if, say J1⊆J2, then x,y∈J2 and since J2 is a filter, x⊓y⊆J2⊆K; if x∈K, then x∈J for some J∈C, thus a⊔x⊆J⊆K; if x⊔y∩K≠∅, then x⊔y∩J≠∅ for some J∈C and then x⊔y⊆J⊆K. Therefore, 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.Corollary 4.
LetF be a proper filter of M. If y∈M such that x∨y≠M for all x∈F. Then, there exists a maximal filter of M containing F and y.In [10], Coquand et al. defined Jacobson’s radical of a proper ideal I of a distributive lattice L as the intersection of maximal ideals of L containing I. The duality filter/ideal allows us to use this definition for filters. After all, a filter of a multilattice M is nothing else than an ideal of the dual of M.Definition 6.
For every proper filterF of M, let KF=M|M is a maximal filter of M containing F. Jacobson’s radical of F is the filter of M denoted JMF and defined as follows: JMF=∩M∈KFM. It is the intersection of all maximal filters of M containing F.Notice thatKF is not empty according to Theorem 4. It is therefore a filter since any intersection of filters is a filter.Theorem 5.
LetF be a proper filter of M. Then,(6)JMF=x∈M|∀y∈M,x∨y=M⇒∃z∈F,z∨y=M.Proof.
For simplicity, let us note(7)L=x∈M|∀y∈M,x∨y=M⇒∃z∈F,z∨y=M.
LetM be a maximal filter of M containing F and let x∈M, x∈L. By Lemma 2, either x∈M or there exists y∈M such that x∨y=M. If x∉M, then x∨y=M. Since x∈L, there exists z∈F such that z∨y=M. Since z,y∈M and M is a filter of M, z∨y⊆M that is M=M. This contradicts the fact that M is proper. Thus necessarily, x∈M and it follows that L⊆JMF.
Suppose thatJMF/L≠∅ and let x∈JMF/L. Then, there exists y∈M, such that x∨y=M and z∨y≠M for all z∈F. Thus, there exists a maximal filter M0 containing F and y. Hence, x,y∈M0, and then x∨y⊆M0 that is M=M0, which contradicts the fact that M0 is a proper filter of M. This implies JMF/L=∅. Therefore, JMF=L and the proof is completed.Theorem5 induces the following two results:Corollary 5.
IfF=a, we write JMa instead of JMa and we have(8)JMa=x∈M|∀y∈M,x∨y=M⇒a∨y=M.The intersection of all maximal filters ofM containing a.In particular, ifF=⊤, then,(9)JM⊤=x∈M|∀y∈M,x∨y=M⇒y=M,is the intersection of all maximal filters of M.Corollary 6.
LetF1, F2 be two filters of M. Then, the following condition holds:(10)F1⊆F2⇒F1⊆JMF1⊆JMF2.Remark 2.
Keeping the same notations as in Proposition2, we can easily verify that JM⊤⊆∩α∈Δ↑α whenever M is a coherent multilattice.Remark 3.
IfM has a bottom ⊥, then from Lemma 1, the set L we use in Theorem 5 can alternatively be defined as follows:(11)L=x∈M|∀y∈M,⊥∈x⊓y⇒∃z∈F,⊥∈z⊓y.Example 2.
The multilatticeM2 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)
ForF∈↑⊤,↑b5,↑b4, JMF=↑a4∩↑b2∩↑c4=↑b4;(ii)
ForF∈↑a13,↑a11,↑a7,↑a4, JMF=↑a4;(iii)
ForF∈↑b3,↑b2, JMF=↑b2;(iv)
ForF∈↑c12,↑c7,↑c4, JMF=↑c4.
One of the fundamental properties of the Jacobson’s radical is the following:Proposition 4.
IfM is coherent and ℱM is distributive, then,(12)JMF1∩F2=JMF1∩JMF2.Proof.
Letx∈JMF1∩JMF2, that is x∈JMF1 and x∈JMF2 and let y∈M such that x∨y=M. Then, there exists z1∈F1 and z2∈F2 such that z1∨y=M=z2∨y. Since M is coherent, there exists z∈M such that z=z1∧z2⊆F1∩F2. Therefore, z∨y=z1∧z2∨y=z1∨y∧z2∨y=M since ℱM is distributive. This implies x∈JMF1∩F2, and we obtain the inclusion JMF1∩JMF2⊆JMF1∩F2. The reverse inclusion holds from Corollary 6 since F1∩F2⊆F1 and F1∩F2⊆F2.The previous results show that whenℱM is a distributive lattice, the binary relation ≼ defined on M by a≼b if and only if JMb⊆JMa 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.



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