# The Prime Filter Theorem for Multilattices

Authors: Daquin Cédric Awouafack; Blaise Blériot Koguep Njionou; Célestin Lélé
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

The aim of this paper is to establish the prime filter theorem for multilattices.


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## Body

## 1. Introduction

Benado’s pioneering work on posets [1] laid the foundation for a new theory called multilattice theory. This theory will be consolidated by several authors including Olga [2] and Hansen [3] who proposed many characterizations of multilattices. Subsequently, the work of Cabrera et al. [4, 5] led to an algebraization of multilattices. Multilattices appear under two inseparable aspects: both as an ordered structure generalizing ordered lattices and as an algebraic hyperstructure generalizing algebraic lattices.If several authors lean towards this theory, it is precisely because of its numerous applications in decision making, in uncertainty modeling, in fuzzy logic, and so on [6]. To further broaden the scope of multilattices, mathematical concepts such as filter/ideal, congruence, and homomorphism [4, 5, 7] will be studied. However, in this new structure, the notion of associativity is replaced by a stronger notion of m-associativity. We thus lose the interest to define the concepts in a canonical way as well as the notion of distributivity is nontrivial [7], which requires more caution when extending some results such as the prime filter theorem.I.P. Cabrera et al. proved in [5] that the set of all filter of a multilattice is a lattice with respect to inclusion. We observe that for some multilattices, the derived filter lattice is distributive. This allows us to use the notion of distributivity in multilattices.The paper is divided into two sections. Section1 concerns the preliminaries: we recall the definitions and results necessarily to understand the paper and we develop some examples to illustrate the proofs. Section 2 brings out the newly established results.


## 2. Preliminaries

LetP,≤ be a poset, and let S⊆P be a nonempty subset of P. An element u∈P is said to be an upper bound of S if it satisfies s≤u for all s∈S. Dually, l∈P is a lower bound of S if it satisfies l≤s for all s∈S.The set of upper bounds (resp., lower bounds) ofS is denoted by Su (resp. Sl). In particular, if S=s, then we write ↑s (resp., ↓s) instead of su (resp., sl). We also use the following notations and definitions: ↑S=∪s∈S↑s and ↓S=∪s∈S↓s.Definition 1 (see [1]).
A multisupremum (resp., multinfimum) ofS is a minimal (resp., maximal) element of Su (resp., Sl).
We will denote byx⊔y (resp., x⊓y) the set of multisuprema (resp., multinfima) of x,y. When x,y has a unique multisupremum (resp., multinfimum), it is denoted by x∨y or supx,y (resp., x∧y or infx,y). We will also use the following notations and definitions: for all nonempty subsets X and Y of M,(1)X⊔Y=∪x,y∈X×Yx⊔y;X⊓Y=∪x,y∈X×Yx⊓y.Definition 2 (see [1]).
LetM;≤ 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 that there exists z∈x⊔y such that z≤aM2:a≤x and a≤y imply that there exists z∈x⊓y such that a≤z
The previous definition gives rise to two hyperoperations which allow us to look at multilattice from an algebraic perspective.(2)⊔:M2⟶2Ma,b↦a⊔band⊓:M2⟶2Ma,b↦a⊓b.
These hyperoperations satisfy the following properties called axioms of multilattices. In [2, 3], many characterizations are proposed.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 (see [5]).
An algebraic multilattice is any tripleM;⊔,⊓ which satisfies the properties AM1-AM5.
Ordered multilattices and algebraic multilattices are connected as follows.Theorem 1.
The following assertions are satisfied:I.
IfM;≤is an ordered multilattice, set(3)x⊔y=multisupx,yandx⊓y=multinfx,yforallx,y∈M.
Then,M;⊔,⊓ is an algebraic multilattice.
II.Conversely ifM;⊔,⊓is a multilattice, set(4)x≤y⇔x⊓y=x⇔x⊔y=y.
Then,M;≤ is an ordered multilattice.
As the two aspects mentioned in Theorem1 are inseparable, it is possible to manipulate multilattices using both the underlined order and the deduced hyperoperations. We can, therefore, use the following three notations to designate multilattice without fear of ambiguity:(5)M=M;≤=M;⊔,⊓=M;≤,⊔,⊓.
We simply writeM assuming that the underlined order and the corresponding hyperoperations are understood.Definition 4.
(1)
Mis said to be acomplete multilatticeif every subset ofMhas at least one multisupremum and at least one multinfimum.(2)
A⊔−full multilattice(resp.,⊓-full multilattice) is a multilattice in whichx⊔y≠∅(resp.,x⊓y≠∅) for allx,y∈M. A multilattice is said to be full if it is both⊔−full and⊓−full.
There are several proposals of the definition of homomorphism between hyperstructures. The most used definition was introduced by Benado in [1].Definition 5 (see [5]).
A maph:M1⟶M2 between multilattices M1,⊔1,⊓1 and M2;⊔2,⊓2 is said to be a homomorphism if(6)ha⊔1b⊆ha⊔2hbandha⊓1b⊆ha⊓2hbforalla,b∈M1.Remark 1.
If the initial multilattice is full, then the condition to be a multilattice homomorphism can be expressed in terms of equalities as follows:(7)ha⊔1b=ha⊔2hb∩hM1andha⊓1b=ha⊔2hb∩hM1foralla,b∈M1.Notation 1.
Any latticeL,∨,∧ induces a multilattice L,∨˜,∧˜ given by x∨˜y=x∨y and x∧˜y=x∧y for all x,y∈L.
In the framework of multilattices, the notions of distributivity and associativity are not quite canonical as stated in [7].Proposition 1 (see [7]).
LetM,⊔,⊓ be a multilattice. Then, the following conditions hold:(1)
If either⊓or⊔is canonically associative, thenMcollapses to a lattice(2)
If⊓(resp.⊔) is canonically distributive to⊔(resp.,⊓), thenMcollapses to a distributive lattice
The notion of filter is studied in [5] in relation to homomorphism and congruences.Definition 6 (see [5]).
A nonempty subsetF of M is said to be a filter if it satisfies the following conditions:F-1:For allx,y∈F,x⊓y⊆F;F-2:For alla∈Mandx∈F,a⊔x⊆F;F-3:For allx,y∈Msuch thatx⊔y∩F≠∅,x⊔y⊆F.Dually, 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⊆II-2:For alla∈Mandx∈I,a⊓x⊆II-3:For allx,y∈Msuch thatx⊓y∩I≠∅,x⊓y⊆INotation 2.
The set of all filters ofM,⊓,⊔ will be denoted by ℱM.Proposition 2 (see [5]).
LetM 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.Example 1.
The following diagrams give two multilattices and their filter lattices.
The smallest filter ofM containing ∅≠X⊆M is the intersection of all filters of M containing X. It is denoted by X, and if X=x, we write x instead of x.


## 3. Main Results

For every nonempty subsetX of M, set(8)X∗≔∪a⊔b|a⊔b∩↑X≠∅,a,b∈M.WhenX=x, we denote x∗ instead of x∗. We shall offer a description of the filter generated by X. We first list some properties of the operator −∗.Proposition 3.
For every nonempty subsetsX and Y of M, the following conditions hold:(i)
X⊆↑X⊆X∗(ii)
X∩Y∗⊆X∗∩Y∗(iii)
X∪Y∗=X∗∪Y∗
We have the following characterization of filters.Lemma 1.
LetF be a nonempty subset of M. Then, F is a filter of M if and only if F⊓F=F and F∗=F.Proof.
Suppose thatF is a filter of M.
Letz∈M. If z∈F⊓F, then z∈x⊓y for some x,y∈F. Since F is a filter of M, x⊓y⊆F and so z∈F. Thus, F⊓F⊆F. If z∈F, then z∈z⊓z⊆F⊓F. Thus, F⊆F⊓F, and it follows that F⊓F=F.
Clearly,F⊆F∗ (see Proposition 3). If z∈F∗, then there is x,y∈M such that z∈x⊔y and x⊔y∩↑F≠∅, but F=↑F (any filter is upward closed) and then x⊔y∩F≠∅, so x⊔y⊆F and consequently z∈F. Therefore, F∗=F.
Conversely, suppose thatF⊓F=F and F∗=F.
Leta,x,y∈M. If x,y∈F, then x⊓y⊆F⊓F=F, and if x∈F, then a⊔x⊆↑F⊆F∗=F; finally, if x⊔y∩F≠∅, then x⊔y⊆F∗=F. Hence, F is a filter of M.
We define a sequenceXnn∈ℕ recursively as follows:(9)X0=X,X1=X∗and∀n≥1,Xn+1=Xn⊓Xn∗.
The following summarizes the main properties of this sequence.Proposition 4.
LetX be a nonempty subset of M. Then, the sequence Xnn∈ℕ is increasing with respect to the inclusion.Proof.
Clearly,Xn⊆Xn⊓Xn and also Xn⊓Xn⊆Xn⊓Xn∗=Xn+1. Therefore, Xn⊆Xn+1.Theorem 2.
LetX be a nonempty subset of M. Then, X=∪n∈ℕXn.Proof.
Leta,x,y∈M. If x∈Xn′ and y∈Xn′′ for some positive integers n′ and n′′ with n′≥n′′, then x,y∈Xn′, and so x⊓y⊆Xn′⊓Xn′⊆Xn′+1; if x∈Xn, n≥0, then a⊔x⊆↑Xn⊆Xn+1; if x⊔y∩Xn≠∅, then x⊔y⊆Xn∗⊆Xn+1. Hence, ∪n∈ℕXn is a filter of M. Clearly, X⊆∪n∈ℕXn (in fact X=X0).
LetF be a filter of M containing X. We will prove by induction that ∪n∈ℕXn⊆F. We have X0=X⊆F. If Xn⊆F, then Xn⊓Xn⊆F⊓F=F and then Xn+1=Xn⊓Xn∗⊆F∗=F (see Lemma 1). Hence, Xn⊆F for all n∈ℕ and it follows that ∪n∈ℕXn⊆F.Example 2.
We consider the multilattices of Figure1. Let XC=a7, XD=b1,b10. We apply Theorem 2 to obtain the filters of C and D, respectively, generated by XC and XD.(10)X0C=a7;X1C=⊤1,a13,a12,a10,a7,a6;X2C=⊤1,a13,a12,a10,a9,a7,a6,a4,a3;X3C=⊤1,a13,a12,a10,a9,a7,a6,a4,a3,a1;XnC=X3C=↑a1,foralln≥3.X0D=b1,b10;X1D=⊤2,b12,a11,a10,b9,b8,b7,b6,b4,b3,b1;XnD=X1D=↑b1for alln≥1;.
Therefore,XC=↑a1 and XD=↑b1.
The next lemma follows directly from the definition of filter.Figure 1
MultilatticesC and D.Lemma 2.
Letx,y∈M. Then, the following assertions hold:(1)
z∈x⊓y⇒z=x∨y;(2)
z∈x⊔y⇒z⊆x∧y;(3)
z,z′∈x⊔y⇒z=z′.
The inclusion of (2) will generally be strict. For instance, in the multilatticeC of Figure 1, we have a6∧a7=↑a1 but a6⊔a7=a12 and a12=↑a12 with ↑a12⊊↑a1.
Figure2 presents the filter lattice of the multilattices C and D of Figure 1. We can easily claim that ℱC is not distributive since it contains a copy of N5 while ℱD is distributive. This allows us to use the following concept of distributivity.Figure 2
Filter latticesℱC and ℱD.Definition 7.
M is said to be a distributive multilattice if FM=ℱM∪∅ is a distributive lattice.Lemma 3.
LetF1,F2∈ℱM. Then, the following conditions hold:(1)
F1∨F2=∪x∈F1,y∈F2x∨y;(2)
F1∧F2=∪x∈F1,y∈F2x∧y.Proof.
For (1), we claim thatA=∪x∈F1,y∈F2x∨y is a filter of M. Let α∈x∨y and β∈x′∨y′, x,x′∈F1×F1, and y,y′∈F2×F2, then α⊓β⊆x∨y∨x′∨y′=z∨z′, where z∈x⊓x′⊆F1 and z′∈y⊓y′⊆F2. If γ∈M, then γ⊔α⊆↑α⊆x∨y, and if γ⊔γ′∩x∨y≠∅, then γ⊓γ′⊆x∨y since x∨y is a filter of M. Therefore, A is a filter of M. It is easy to see that F1⊆A and F2⊆A, and every filter of M containing F1∪F2 necessarily contains A.
For (2), letℬ=∪x∈F1,y∈F2x∧y. We claim that ℬ is a filter of M. Let α∈x∧y and β∈x′∧y′, x,x′∈F1×F1, and y,y′∈F2×F2, then α⊓β⊆x∧y∨x′∧y′⊆z∧z′, where z∈x⊓x′⊆I and. If γ∈M, then γ⊔α⊆↑α⊆x∧y, and if γ⊔γ′∩x∧y≠∅, then γ⊔γ′⊆x∧y since x∧y is a filter of M. Therefore, ℬ is a filter of M. For all x∈F1 and for all y∈F2, we have x∧y⊆x⊆F1 and x∧y⊆y⊆F2; thus, x∧y⊆F1∧F2. Therefore, ℬ⊆F1∧F2. It obvious that F1∧F2⊆ℬ. In fact, x∈F1∧F2 implies x∈F1 and x∈F2 and then we write x∈x∧x⊆ℬ.Definition 8.
A proper filterP of M is said to be prime if for all F1,F2∈ℱM, F1∩F2⊆P implies F1⊆P or F2⊆P.
From (2) of Lemma3, we deduce the following result.Corollary 1.
LetP be a filter of M. Then, the following conditions are equivalent:(1)
Pis a prime filter(2)
For allx,y∈M,x∧y⊆Pimpliesx∈Pory∈PProof.
The first implication follows directly from Definition8. For the converse, suppose that F1∧F2⊆P but F1⊈P and F2⊈P. Let x∈F1∖P and y∈F2∖P. Then, neither x∈P nor y∈P. From (2), we have x∧y⊈P. However, from Lemma 3, x∧y⊆F1∩F2⊆P which is a contradiction.
Letγ:M⟶ℱM, x↦x.Theorem 3.
LetP⊆M be a nonempty subset of M. Then, P is a prime filter of M if and only if there exists a lattice homomorphism f from ℱM to the lattice 2=0,1;min,max such that f°γ−10=P.Proof.
Suppose thatP is prime, then DsP=F∈ℱM|F⊆P is also prime in ℱM and then there is a lattice homomorphism f from ℱM to the lattice 2 such that DsP=f−10. However, we observe that γ−1DsP=P. It follows that f°γ−10=P.
Conversely, iff is a lattice homomorphism from ℱM to 2, then f−10 is a prime filter of ℱM. This implies f°γ−10=γ−1f−10 is a prime filter of M.Corollary 2.
LetP a filter of M. Then, P is a prime filter if and only if there exists a multilattice homomorphism φ from M to 2 such that φ−10=P.Proof.
Letf be an homomorphism form ℱM to 2. Let x,y∈M, then on the one hand, we have f°γx⊔y=fx⊔y=fx∨y=fx∨˜fy since x⊔y=x∨y. Hence, f°γx⊔y=f°γx∨˜f°γy. On the other hand, we have f°γx⊓y=fx⊓y⊆fx∧y=fx∧˜fy since x⊓y⊆x∧y which implies f°γx⊓y⊆f°γx∧˜f°γy. This proves that f°γ is a multilattice homomorphism. Then, we use Theorem 3 to obtain the existence of φ.Lemma 4.
LetF1 and F2 be two filters of M, and let I be an ideal of M. Then,(11)I∩F1≠∅andI∩F2≠∅⇒F1∧F2∩I≠∅.Proof.
Letx∈I∩F1 and y∈I∩F2. Then, firstly, x⊔y⊆I since I is an ideal of M; secondly, x⊔y⊆↑x⊆F1 and also x⊔y⊆↑y⊆F2. Thus, x⊔y⊆F1∧F2. Therefore, ∅≠x⊔y⊆F1∧F2∩I.
A. Rahanamai–Barghi established the prime ideal theorem for meet-hyperlattices [8] and for distributive hyperlattices [9]. In the following theorem, we state and prove the prime filter theorem for multilattices.Theorem 4 (prime filter theorem).
LetM be a distributive multilattice. Let F be a proper filter of M and I a proper ideal of M such that F∩I=∅. Then, there exists a prime filter P of M such that F⊆P and P∩I=∅.Proof.
LetX be the set of all filters of M containing F and disjoint from I.(12)X=G|F⊆G⊆M,Gis a filter,G∩I=∅.
We claim thatX satisfies Zorn’s lemma. Since F∈X, the set X is not empty. Let C be a chain in X, and let N=∪X∈CX. Let x,y∈M. Firstly, if x,y∈N, then x∈X1 and y∈X2 for some X1, X2∈C; since C is a chain, either X1⊆X2 or X2⊆X1. If X1⊆X2, then x,y∈X2 and so x⊓y⊆X2⊆N. Secondly, if x∈M and y∈N, then y∈Y for some Y∈C; since Y is a filter, x⊔y⊆Y⊆N. Thirdly, if x⊔y∩N≠∅, then x⊔y∩Z≠∅ for some Z∈C; since Z is a filter, x⊔y⊆Z⊆N. Thus, N is a filter of M. It is obvious that F⊆N and N∩I=∅, verifying that N∈X. Therefore, Zorn’s lemma X has a maximal element, say P. We claim that P is prime. Indeed suppose to the contrary that P is not prime. Then, there exists F1,F2∈ℱM such that F1∧F2⊆P, F1⊈P, and F2⊈P. P∨F1 and P∨F2 both are filters which strictly contain P. Because of the maximality of P, clearly P∨F1∩I≠∅ and P∨F2∩I≠∅. Thus, from Lemma 4, P∨F1∧P∨F2∩I≠∅. Since ℱM is distributive, we have P∨F1∧P∨F2=P∨F1∧F2=P. This implies P∩I≠∅, a contradiction.Corollary 3.
LetM,⊔,⊓,⊤ be a distributive multilattice with top element, ⊤. If I is a proper ideal of M, then there exists a prime filter P such that P∩I=∅.Proof.
SinceI is proper, we have ⊤∉I. Hence, we apply Theorem 4 with F=⊤ to obtain the desired conclusion.


## 4. Conclusion

We have established the prime filter theorem and proposed some characterizations.



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