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The poset of closure systems on an infinite poset:
Detachability and semimodularity
Christian Ronse
To cite this version:
Christian Ronse. The poset of closure systems on an infinite poset: Detachability and semimodularity.
Portugaliae Mathematica, European Mathematical Society Publishing House, 2010, 67 (4), pp.437-452.
�10.4171/PM/1872�. �hal-02882874�
The poset of closure systems on an infinite poset:
detachability and semimodularity
Christian Ronse
Abstract. Closure operators on a poset can be characterized by the corresponding clo- sure systems. It is known that in a directed complete partial order (DCPO), in particular in any finite poset, the collection of all closure systems is closed under arbitrary inter- section and has a “detachability” or “anti-matroid” property, which implies that the collection of all closure systems is a lower semimodular complete lattice (and dually, the closure operators form an upper semimodular complete lattice).
After reviewing the history of the problem, we generalize these results to the case of an infinite poset where closure systems do not necessarily constitute a complete lattice;
thus the notions of lower semimodularity and detachability are extended accordingly. We also give several examples showing that many properties of closure systems on a complete lattice do not extend to infinite posets.
Mathematics Subject Classification (2000). Primary 06A15; Secondary 06C10.
Keywords. Poset, closure operator, closure system, MLB-closed set, detachability, lower semimodularity.
1. Introduction: history of the problem
LetP be a partially ordered set (from now onposet). Following [2], for anyx∈P, let us write
x↑={y∈P |y≥x} and x↓={y∈P|y≤x} .
ForX ⊆P, we say thatX is adown-set if∀x∈X,x↓⊆X [2]. Following [14], we call anoperator on P a mapP →P, and forx∈P and an operatorψ onP, the image ofxbyψ is writtenψ(x); given two operatorsξ and ψon P the operator P →P:x7→ψ(ξ(x)) is thecompositionofξfollowed byψ, following [14] we write itψξrather than the usualψ◦ξ(as in [2]).
A closure operator onP is an operatorϕthat is isotone (x≤y ⇒ ϕ(x) ≤ ϕ(y)),extensive(x≤ϕ(x)) andidempotent(ϕ(ϕ(x)) =ϕ(x)). Equivalently [9, 20]:
∀x, y∈P, x≤ϕ(y) ⇐⇒ ϕ(x)≤ϕ(y) . For any closure operatorϕ, let
Inv(ϕ) ={x∈P |ϕ(x) =x}={ϕ(x)|x∈P}
be theinvariance domain of ϕ [14] (it is also called the range ofϕ [12]). Let us write Φ(P) for the set of all closure operators on P.
A closure system on P is a subset S of P such that for any x ∈ P, x↑∩S is non-void and has a least element. In [12] such a set is called a closure range.
Let us write Σ(P) for the set of closure systems onP. It has been known at least since [18] that there is a bijection between Φ(P) and Σ(P): to a closure operator ϕwe associate the closure systemInv(ϕ), and conversely to a closure systemS we associate the closure operator mapping eachx∈P to the least element ofx↑∩S.
Closure systems can be ordered by set inclusion, while closure operators can be ordered elementwise: ϕ1 ≤ϕ2 iff ∀x∈P, ϕ1(x)≤ϕ2(x). It is easily shown (cf.
[14]) that two closure operatorsϕ1 andϕ2satisfy
ϕ1≤ϕ2 ⇐⇒ Inv(ϕ1)⊇Inv(ϕ2) ⇐⇒ ϕ2ϕ1=ϕ2 ⇐⇒ ϕ1ϕ2=ϕ2 . (1) Thus the bijectionϕ7→ Inv(ϕ) between the posets Φ(P) and Σ(P) is a dual iso- morphism.
A classical result of Ward [24] shows that in case P is a complete lattice (whose universal bounds are written 0 and 1), a subset S of P is a closure sys- tem iff it is closed under arbitrary infima (in particular for the empty infimum, 1 ∈ S). Furthermore, Φ(P) and Σ(P) are dually isomorphic complete lattices.
More precisely, Φ(P) is closed under elementwise infimum: forϕi ∈Φ(P) (i∈I), V
i∈Iϕi : x 7→ V
i∈Iϕi(x) belongs to Φ(P) (in particular, for I = ∅, x 7→ 1 is the greatest closure operator); on the other hand, Σ(P) is closed under arbitrary intersection: forSi ∈Σ(P) (i∈I), T
i∈ISi∈Σ(P) (in particular, forI=∅,P is the greatest closure system).
Given a complete latticeP (with greatest element1),{1}is the least element of Σ(P), and the{1, x}(x∈P\{1}) are theatomsof Σ(P). Thus everyS∈Σ(P) is a union of atoms, and the complete lattice Σ(P) isatomistic.
Manara [17] showed that for a complete lattice P, given two closure systems S1 andS2 on P such thatS1 ⊂S2, in the lattice Σ(P) we haveS2≻S1 (i.e.,S2
coversS1) iff S2\S1 is a singleton. It follows then that the lattice Σ(P) is lower semimodular:
∀S, S′ ∈Σ(P), S∨S′≻S′ =⇒ S≻S∩S′ ; hence dually the lattice Φ(P) is upper semimodular:
∀ϕ, ϕ′∈Φ(P), ϕ′≻ϕ∧ϕ′ =⇒ ϕ∨ϕ′ ≻ϕ .
(Here ϕ∨ϕ′ denotes the join in the lattice Φ(P), not the elementwise join x7→
ϕ(x)∨ϕ′(x)). As remarked in [4], Ore [21] had shown these two results in the case where the complete latticeP isP(E), the Boolean lattice of all subsets of a setE.
Note that [17] contains several erroneous statements, arising mainly because of a misquote of previous results by Dwinger [10, 11]: Manara confused the notion of a “well-ordered complete lattice” (i.e., a successor ordinal) with that of a “totally ordered complete lattice” (i.e., a complete chain). For example [10] showed that Φ(P) is distributive iff P is a complete chain, and that Φ(P) is Boolean iff P is a successor ordinal; however Theorem 1.12 of [17] states (citing [11]) that Φ(P) is distributive iff it is Boolean, iffP is a complete chain.
What can we say in casePis not a complete lattice ? Several authors [1, 19, 13]
presented proofs that if the posetP satisfies theACC (ascending chain condition, or equivalently, every directed subset of P has a greatest element), then Σ(P) is closed under arbitrary intersection (and hasP as greatest element), hence it is a complete lattice. This holds in particular ifP is finite. The two proofs in [13] are correct. We have not checked the correctness of the proof in [1], but [23] showed that the argument of [19] is flawed. The result can in fact be obtained by a simple argument. Letϕi (i∈I) be closure operators with associated closure systemsSi; letM be the monoid generated by theϕi (i∈I); then for anyx∈P, the set of µ(x) for µ∈ M is directed (forµ, µ′ ∈ M, µµ′ ∈ M and µ(x), µ′(x)≤ µµ′(x)), hence it has a greatest elementψ(x); it is easily seen thatψ(x) is the least element of x↑∩T
i∈ISi, hence T
i∈ISi is a closure system; then ψ is a closure operator, namely the supremum in Φ(P) of theϕi (i∈I).
The second proof in [13] relies on a generalization of Ward’s [24] characteriza- tion of a closure system on a complete lattice as a subset closed under arbitrary infima (and containing the greatest element1). A subsetS of a poset P is called MLB-closed [22] iff for anyX ⊆S, every maximal lower bound ofX must belong toS (in particular forX =∅, every maximal element ofP must belong toS). It is easily seen that every closure system is MLB-closed, and that the collection of all MLB-closed subsets of P is closed under arbitrary intersection (in particular for the empty intersection: P is MLB-closed). Now [13] showed that in a posetP satisfying the ACC, a subsetS ofP is a closure system iff it is MLB-closed, hence the collection Σ(P) of all closure systems is closed under arbitrary intersection.
In [22] the equivalence between MLB-closed sets and closure systems was ex- tended to the case whereP is a DCPO (a poset where every directed subset has a supremum). It was then indicated that it is also possible to prove directly that Σ(P) is closed under arbitrary intersection. Indeed, if we return to the above ar- gument with the monoidMgenerated by the closure operatorsϕi (i∈I), for any x∈P, the directed set of allµ(x), forµ∈M, has a supremumψ(x). This gives an extensive and isotone operatorψ, and the least closure operatorϕ≥ψcan be con- structed by several methods, see [6], in particular by transfinite induction, as did [16] in the case wherePis a complete lattice. Then we haveInv(ϕ) =T
i∈IInv(ϕi).
Manara’s [17] results about covers and semimodularity extend also to any poset Pwhere Σ(P) is closed under arbitrary intersection (in particular ifP is a DCPO).
For any X ⊆ P, the set of all S ∈ Σ(P) such that X ⊆ S, is closed under intersection, hence there is a least closure system containingX, we write itσ(X) and call it theclosure system generated by X. In [13] it was shown that if P has ACC, then Σ(P) satisfies theanti-matroid exchange property:
∀S∈Σ(P), ∀x, y∈P\S, x6=y, y∈σ(S∪ {x}) =⇒ x /∈σ(S∪ {y}) . (2) Ern´e [12] showed that for any posetP with Σ(P) closed under arbitrary intersec- tion, Σ(P) isdetachable:
∀S∈Σ(P), ∀x∈P\S, σ(S∪ {x})\ {x} ∈Σ(P) . (3) This concept comes from [15] (where one saysextremally detachable), and obviously detachability (3) implies the anti-matroid exchange property (2); in [15] conditions are given under which both properties are equivalent. From any of (2,3) follow Manara’s result stating that forS1, S2∈Σ(P) such thatS1⊂S2, S2coversS1 in the lattice Σ(P) iffS2\S1 is a singleton, hence that Σ(P) is lower semimodular.
In case the poset P is not a DCPO, Σ(P) does not necessarily constitute a complete lattice, and even in case Σ(P) is a complete lattice, the infimum operation in Σ(P) is not necessarily the intersection, see Section 3.
The goal of this paper is to generalize the notions of semimodularity and de- tachability to the case where Σ(P) is not necessarily a complete lattice, and to prove the corresponding extensions of Manara’s [17] and Ern´e’s [12] results. We obtain the following detachability property:
∀S0, S1∈Σ(P), S0⊂S1, ∀x∈S1\S0,
∃S2∈Σ(P), S0∪ {x} ⊆S2⊆S1, S2\ {x} ∈Σ(P) . (4) It is easily seen that if Σ(P) is closed under arbitrary intersection, then (4) is equivalent to (3). Next we show that the poset Σ(P) is lower semimodular in the following sense:
∀S1, S2∈Σ(P), if in Σ(P) the joinS1∨S2 exists andS1∨S2≻S1, then the meetS1∧S2exists and S2≻S1∧S2.
Note that in this case we will have S1∨S2 = S1∪S2 and S1∧S2 = S1∩S2. By duality, we obtain that the poset Φ(P) is upper semimodular in the following sense:
∀ϕ1, ϕ2∈Φ(P), if in Φ(P) the meetϕ1∧ϕ2exists andϕ1≻ϕ1∧ϕ2, then the joinϕ1∨ϕ2exists and ϕ1∨ϕ2≻ϕ2.
Here we have no indication on the explicit values of (ϕ1∧ϕ2)(x) and (ϕ1∨ϕ2)(x) forx∈P.
Concerning the other properties of Σ(P) in the case where P is a complete lattice (e.g., Σ(P) is atomistic, Σ(P) is closed under arbitrary intersection, closure systems coincide with MLB-closed sets, etc.), we will see in Section 3 that they are generally lost in the case whereP is an arbitrary infinite poset. However, some of these properties may hold if one makes appropriate assumptions onP.
2. The main argument
In this section we shall fix a posetP. At the basis of detachability is the following generalization of some constructions made in [19]:
Proposition 2.1. Let S0, S1∈Σ(P)such that S0⊆S1, and let A be a down-set in P. Let S2 = (S0\A)⊎(S1∩A). Then S2 =S0∪(S1∩A) = (S0∪A)∩S1, S0⊆S2⊆S1 andS2∈Σ(P).
The closure operators ϕ0, ϕ1, ϕ2 corresponding to S0, S1, S2 satisfy ϕ1≤ϕ2 ≤ ϕ0, and for every x∈P we have
ϕ2(x) =
(ϕ1(x) ifϕ1(x)∈A , ϕ0(x) ifϕ1(x)∈/A .
Proof. AsS0⊆S1, we getS2= (S0\A)∪(S0∩A)∪(S1∩A) =S0∪(S1∩A); by the modular equality, we obtainS2 = (S0∪A)∩S1, andS0⊆S2⊆S1. For any x∈P we have two cases:
(1) ϕ1(x) ∈ A. Then ϕ1(x) ∈ S1∩A ⊆ S2. As ϕ1(x) is the least element of x↑∩S1 andϕ1(x)∈x↑∩S2, withx↑∩S2⊆x↑∩S1, we deduce thatϕ1(x) is the least element ofx↑∩S2. Setϕ2(x) =ϕ1(x).
(2) ϕ1(x)∈/ A. For anyy∈x↑∩S1, we haveϕ1(x)≤y, and asAis a down-set and ϕ1(x) ∈/ A, we get y /∈ A. Hence x↑∩S1∩A = ∅, and as S0 ⊆ S1, x↑∩S0∩A=∅. Hence
x↑∩S0= (x↑∩S0\A)∪(x↑∩S0∩A) =x↑∩S0\A
= (x↑∩S0\A)∪(x↑∩S1∩A) =x↑∩S2 .
Nowϕ0(x) is the least element ofx↑∩S0=x↑∩S2. Setϕ2(x) =ϕ0(x).
Thus in both casesϕ2(x) is the least element ofx↑∩S2, soS2is a closure system and ϕ2is a closure operator. We haveϕ2(x) =ϕ1(x) forϕ1(x)∈A, andϕ2(x) =ϕ0(x) forϕ1(x)∈/A.
An interesting particular case is forS1=P(that is,ϕ1is the identity operator):
Corollary 2.2. Given a closure system S on P and a down-set A in P, then S∪Ais a closure system. Ifϕis the closure operator corresponding toS, then the closure operatorϕA corresponding toS∪Ais given by
ϕA(x) =
(x forx∈A , ϕ(x) forx /∈A .
We obtain from Proposition 2.1 the detachability condition (4):
Corollary 2.3. For anyS0, S1∈Σ(P)such thatS0⊂S1and for anyx∈S1\S0, there existsS2∈Σ(P)such that S0∪ {x} ⊆S2⊆S1 andS2\ {x} ∈Σ(P).
Proof. A =x↓ and A′ = x↓\ {x} = {y ∈ P | y < x} are down-sets. Let S2 = (S0\A)⊎(S1∩A) andS2′ = (S0\A′)⊎(S1∩A′). By Proposition 2.1,S2, S2′ ∈Σ(P), S2=S0∪(S1∩A),S2′ =S0∪(S1∩A′),S2⊆S1andS0⊆S2′; asx∈S1∩A, we have x∈S2, and asx /∈S0,S2\ {x}= (S0\ {x})∪(S1∩A\ {x}) =S0∪(S1∩A′) =S2′. HenceS0∪ {x} ⊆S2⊆S1 andS2′ =S2\ {x}, withS2, S2′ ∈Σ(P).
This gives thus a generalization of Manara’s first result:
Corollary 2.4. ForS0, S1∈Σ(P)such thatS0⊂S1, we haveS1≻S0(inΣ(P)) iffS1\S0 is a singleton.
Proof. Take x ∈ S1 \S0, then by Corollary 2.3 there is S2 ∈ Σ(P) such that S2\ {x} ∈ Σ(P) and S0 ⊆ S2\ {x} ⊂ S2 ⊆ S1. If S1 covers S0 in Σ(P), we must necessarily have S2\ {x} =S0 and S2 = S1; hence S1\S0 is a singleton.
Conversely, if S1\S0 is a singleton, then S1 covers S0 in P(E), so obviously it coversS0 in the smaller poset Σ(P).
It follows by induction that for S0, S1∈Σ(P) such thatS0⊂S1, the interval [S0, S1] in Σ(P) has finite height iffS1\S0is finite, and then the height of [S0, S1] in Σ(P) equals the size of S1\S0.
Returning to Corollary 2.3, writingS3=S2\ {x}, we obtain thus:
∀S0, S1∈Σ(P), S0⊂S1 =⇒ ∃S2, S3∈Σ(P), S0⊆S3≺S2⊆S1 . In other words every interval in Σ(P) contains a cover; one says then that Σ(P) is weakly atomic[6].
In order to prove the lower semimodularity of Σ(P), previous works [17, 13, 12]
required Σ(P) to be closed under arbitrary intersection; since we do not make this assumption, we need to show that in some particular cases the intersection of two closure systems is a closure system:
Proposition 2.5. Let S1, S2 ∈Σ(P) such that S2\S1 is finite; then S1∩S2 ∈ Σ(P).
Givenϕ1, ϕ2the closure operators corresponding toS1, S2, the closure operator corresponding toS1∩S2 is(ϕ2ϕ1)nϕ2, wheren=|S2\S1|.
Proof. Letn=|S2\S1|. For anyx∈P,ϕ2(x)∈S2. For anyy∈S2, we have two cases:
(1) Ify∈S1∩S2, thenϕ1(y) =y=ϕ2(y), soϕ2ϕ1(y) =y.
(2) Ify ∈ S2\S1, then y /∈ Inv(ϕ1), so y < ϕ1(y), and ϕ1(y)≤ϕ2ϕ1(y), thus y < ϕ2ϕ1(y).
Letx∈ P; for any t ∈ N, (ϕ2ϕ1)tϕ2(x)∈ S2. If there is somet < n such that (ϕ2ϕ1)tϕ2(x)∈S1∩S2, case 1 yields that
(ϕ2ϕ1)t+1ϕ2(x) =ϕ2ϕ1 (ϕ2ϕ1)tϕ2(x)
= (ϕ2ϕ1)tϕ2(x) ,
and thus (ϕ2ϕ1)nϕ2(x)∈S1∩S2. If (ϕ2ϕ1)tϕ2(x)∈S2\S1 for everyt < n, by case 2 we get
ϕ2(x)<· · ·<(ϕ2ϕ1)tϕ2(x)<· · ·<(ϕ2ϕ1)nϕ2(x) , with ϕ2(x), . . . ,(ϕ2ϕ1)n−1ϕ2(x)∈S2\S1 ;
as|S2\S1|=n, this means that (ϕ2ϕ1)nϕ2(x)∈/ S2\S1, that is (ϕ2ϕ1)nϕ2(x)∈ S1∩S2. Thus we have shown that for any x∈P, (ϕ2ϕ1)nϕ2(x)∈S1∩S2. Now for x∈ S1∩S2, by induction case 1 gives (ϕ2ϕ1)nϕ2(x) = x; hence (ϕ2ϕ1)nϕ2
is idempotent. Clearly (ϕ2ϕ1)nϕ2 inherits the isotony and extensivity ofϕ1 and ϕ2. Therefore (ϕ2ϕ1)nϕ2is a closure operator, withInv (ϕ2ϕ1)nϕ2
=S1∩S2, so S1∩S2 is a closure system.
We obtain thus the lower semimodularity of Σ(P), the following argument is the same as in previous works [17, 13, 12]:
Corollary 2.6. ForS1, S2∈Σ(P), if inΣ(P)the joinS1∨S2exists andS1∨S2≻ S1, then the meetS1∧S2exists andS2≻S1∧S2. We have thenS1∨S2=S1∪S2
andS1∧S2=S1∩S2.
Proof. Suppose thatS1∨S2 exists andS1∨S2≻S1; thusS1⊆S1∪S2⊆S1∨S2. We may not haveS1∪S2=S1, otherwiseS2⊆S1 and thenS1∨S2=S1. Hence S1⊂S1∪S2 ⊆S1∨S2, and by Corollary 2.4, (S1∨S2)\S1 is a singleton; thus we getS1∨S2=S1∪S2. Now (S1∨S2)\S1= (S1∪S2)\S1=S2\S1; as this is a singleton, Proposition 2.5 givesS1∩S2∈Σ(P). Then obviously S1∩S2 is the meet ofS1andS2in Σ(P). We haveS2\(S1∩S2) =S2\S1; as this is a singleton, S2≻S1∩S2 by Corollary 2.4.
By duality, Φ(P) is upper semimodular in the following sense: for ϕ1, ϕ2 ∈ Φ(P), if in Φ(P) the meetϕ1∧ϕ2 exists andϕ1≻ϕ1∧ϕ2, then the joinϕ1∨ϕ2
exists andϕ1∨ϕ2≻ϕ2.
3. Some counterexamples to other properties
We have shown that the results of Ern´e and Manara, namely that Σ(P) is detach- able and lower semimodular, can be extended to the case whereP is an arbitrary poset. What about the other properties satisfied by Σ(P) in caseP is a complete lattice ? It is easy to show that the following hold:
(Pa) Every closure system is MLB-closed, and the collection of all MLB-closed sets is closed under arbitrary intersections.
(Pb) If the poset P has a greatest element 1, then Σ(P) has least element {1}
and atoms{1, x}for allx∈P\ {1}, so it is atomistic.
(Pc) IfP is a meet-semilattice (i.e., every two elements of P have a meet), then for any S1, S2 ∈ Σ(P), the joinS1∨S2 exists in Σ(P), it is the set of all x1∧x2 for x1 ∈S1 andx2 ∈S2 (NB: elements of S1∪S2 take this form:
∀x1∈S1,x1=x1∧ϕ2(x1), where ϕ2(x1)∈S2); in particular, ifS1andS2
are finite, thenS1∨S2is finite.
(Pd) IfP is a complete meet-semilattice (i.e., every non-void subset of P has an infimum), then for any non-void family Si ∈ Σ(P) (i ∈ I), its supremum W
i∈ISi exists in Σ(P), it is the set of V
i∈Ixi for all choices xi ∈ Si; in particular, Σ(P) is a complete join-semilattice (i.e., every non-void subset of P has a supremum).
(Pe) If P = P(E), the Boolean lattice of all subsets of a finite set E, then [5]
Σ(P) isjoin-semidistributive:
∀X, Y, Z∈Σ(P), ifX∨Y =X∨Z, thenX∨Y =X∨(Y ∧Z) ; hence it isjoin-pseudocomplemented:
∀X ∈Σ(P),
Y ∈Σ(P)|X∨Y =P has a least element.
However, we will see through counterexamples that in the general case of an arbi- trary posetP, none of these properties can be extended. More precisely:
(F1) Σ(P) does not necessarily constitute a complete lattice, nor even a lattice.
(F2) Even in case Σ(P) is a complete lattice, the infimum operation in Σ(P) is not necessarily the intersection.
(F3) Even in case Σ(P) is a complete lattice with the infimum operation given by the intersection, Σ(P) can be larger than the collection of all MLB-closed sets.
(F4) In caseP does not have a greatest element, Σ(P) can have elements that are not joins of atoms; in some cases it may have no atoms.
(F5) There can be finite S1, S2 ∈Σ(P) with an infinite joinS1∨S2 in Σ(P), or even having no join in Σ(P).
(F6) Even in case Σ(P) is a complete lattice, it can be neither join-semidistributive nor join-pseudocomplemented.
These failures of expected properties are shown through several counterexamples.
The first ones are posets without greatest element:
A. Naturals: Let P = N, the set of natural integers. Every subset of N (in- cluding ∅) is MLB-closed; on the other hand a subset ofN is a closure system if and only if it is infinite. Then Σ(N) is closed under non-void unions and has N as greatest element; it is thus a complete join-semilattice. However two closure systems onN do not necessarily have a meet in Σ(N); for example letS0= 2N (the set of even naturals) andS1= 2N+1 (the set of odd naturals), asS0∩S1=∅,
there is no closure system contained in both S0 and S1; seen otherwise, S0 and S1 correspond to the two closure operators ϕ0 andϕ1 that map every natural to the least even (resp., odd) natural above it, and iteratingϕ0ϕ1on anyn∈N, the sequence (ϕ0ϕ1)t(n) (t∈N) goes to infinity. Hence Σ(N) is not a lattice, cf. (F1).
Furthermore, there are no atoms nor any least element in Σ(N), cf. (F4).
(a) a
c d
n b
2 1 0
(b)
ε
0 1
00 01 10 11
000 001 010 011 100 101 110 111
(c) a
b c
n
2 1 0
(d) b c
a
A
Figure 1. Hasse diagrams of posets: (a) fence over naturals; (b) 01-words; (c) hat over naturals; (d) hat over antichain.
B. Fence over naturals: This example comes from [19, 22]. Let P = N∪ {a, b, c, d}, where {a, b, c, d} is a fence (a > c, b > c, b > d) standing above N (∀n ∈ N, c, d > n), with the usual order on N; the Hasse diagram of P is shown in Figure 1 (a). The MLB-closed subsets of P are all those containing {a, b, c}. On the other hand, the closure systems onP can be of two forms: either
{a, b, c} ∪X for anyX ⊆N, or{a, b, c, d} ∪Y for aninfinite Y ⊆N. It is easily checked that Σ(P) is closed under non-void unions and constitutes a complete lattice with universal bounds{a, b, c}andP. However givenS0={a, b, c, d} ∪2N andS1={a, b, c, d} ∪(2N+ 1), we haveS0, S1∈Σ(P), withS0∩S1={a, b, c, d}, but in Σ(P) we haveS0∧S1={a, b, c}. Thus the infimum operation in Σ(P) is not the intersection, cf. (F2). Moreover, as there is no least closure system containing {a, b, c, d}, the operator σ is not defined and the properties (2,3) make no sense here. The atoms of Σ(P) are all {a, b, c, n} for n ∈N, thus joins of atoms have the form {a, b, c} ∪X for X ⊆ N; hence Σ(P) is not atomistic, cf. (F4); it is however atomic (i.e., every nonzero element of Σ(P) contains an atom). Finally, letS2 ={a, b, c} ∪N =P \ {d}; then S2 ∈ Σ(P), S2∨S0 = S2∨S1 = P, but S2∨(S0∧S1) =S2∨ {a, b, c}=S2; in fact there is no least element in the set of allS∈Σ(P) such thatS2∨S =P. Thus Σ(P) is neither join-semidistributive nor join-pseudocomplemented, cf. (F6).
C. 01-words: Let P be the set of words on the alphabet {0,1}, with prefix ordering, see the Hasse diagram in Figure 1 (b); equivalently, P =S
n∈N{0,1}n, where forx= (x0, . . . , xm−1)∈ {0,1}m andy = (y0, . . . , yn−1)∈ {0,1}n we have x≤y iff m ≤n and xi =yi fori = 0, . . . , m−1. Then P is a complete meet- semilattice; the least element is the empty word. It is easily checked that the only closure operator onP is the identity, thusP is the only closure system and Σ(P) is trivially closed under arbitrary intersection. On the other hand the MLB-closed sets are all sets closed under non-empty infimum, there are infinitely many such sets, cf. (F3).
Our next counterexamples are posets having a greatest element; we will see that this property does not bring much more than the fact that every closure system is a join of atoms (cf. property (Pb) above):
D. Hat over naturals: This example comes from [19]. Let P =N∪ {a, b, c}, where {a, b, c} is a hat (a > b, a > c) standing above N (∀n ∈ N, b, c > n), with the usual order onN; the Hasse diagram ofP is shown in Figure 1 (c). The MLB-closed subsets ofP are all those containinga. On the other hand, the closure systems onP are the sets{a} ∪X,{a, b} ∪X and{a, c} ∪X for anyX ⊆N, and the sets {a, b, c} ∪Y for an infinite Y ⊆N. Now given S0 = {a, b, c} ∪2N and S1={a, b, c} ∪(2N+ 1), we haveS0, S1∈Σ(P), withS0∩S1={a, b, c}∈/ Σ(P), so in Σ(P) the pair{S0, S1}has two maximal lower bounds{a, b}and{a, c}. Thus Σ(P) is not a lattice, cf. (F1). Note that both{a, b}and{a, c}are finite elements of Σ(P), but they have no join in Σ(P), cf. (F5).
E. Hat over antichain: LetP =A∪ {a, b, c}, whereA is an infinite antichain (∀x, y∈A,x6< yandy6< x) and{a, b, c}is a hat (a > b,a > c) standing aboveA (∀x∈A,b, c > x); the Hasse diagram ofP is shown in Figure 1 (d). SinceP has finite height, it satisfies the ACC condition, thus closure systems on P coincide with MLB-closed subsets of P [13], they are: the sets {a} ∪X, {a, b} ∪X and {a, c} ∪X for anyX ⊆A, and P. Note that the two pairs {a, b} and {a, c} are atoms, with join{a, b} ∨ {a, c}=P which is infinite, cf. (F5).
F. Finite/co-finite: LetPbe the collection consisting of all finite and co-finite (complement of finite) subsets ofN. ThenP, ordered by the inclusion relation, is a Boolean lattice, but not a complete lattice. Note that a subset Q of P is MLB-closed iff for any familySi ∈Q (i∈I), eitherT
i∈ISi∈Q orT
i∈ISi ∈/ P (i.e., T
i∈ISi is infinite with infinite complement); in other words Qis the trace onPof a closure system (Moore family) onP(E). Now Σ(P) is not a lattice, cf.
(F1). Indeed, for anyn∈N, letAn= 2N∩[0, n] ={k∈2N| k≤n}, and define S0 ⊆ P (resp., S1 ⊆ P) to be the set of all co-finite subsets B of N such that 2N⊆B, and of allAn∪X forn∈4N (resp.,n∈4N+ 2) andX a finite subset of 2N+ 1; we can check thatS0,S1∈Σ(P), butS0andS1 have no meet in Σ(P), becauseS0∩S1is the set of of all co-finite subsetsBofNcontaining 2N, and the closure systems contained in it are its finite parts closed under intersection, and none of them is maximal.
4. Discussion and conclusion
Our counter-examples in Section 3 indicate that in the general case, one probably cannot obtain more than the results given in Section 2. Nevertheless, in view of previous works [17, 13, 22, 12, 5], we can inquire about the most general properties that a posetP should satisfy in order to guarantee some properties for Σ(P); thus we can raise the following questions:
• What are the conditions for Σ(P) to be a complete lattice ? to be closed under arbitrary intersection ? to coincide with the collection of all MLB- closed subsets ofP ?
• IfP does not have a greatest element, when is Σ(P) an atomistic complete lattice ?
• We showed that forS1, S2∈Σ(P) such thatS2\S1is finite,S1∩S2∈Σ(P).
What are then the conditions for the joinS1∨S2 to exist in Σ(P) ?
• What are the conditions for Σ(P) to be join-semidistributive ? to be join- pseudocomplemented ?
Another interesting question is the analysis of the properties of Σ(P) in case P has a greatest element (cf. property (Pb) at the beginning of Section 3). It seems that adding a greatest element to a poset increases considerably the collection of all closure systems.
Let us now briefly discuss some related questions concerning closure operators and systems, in particular their characterizations.
IfP =P(E), the Boolean lattice of all subsets of a finite setE, the fact that Σ(P) is join-semidistributive and join-pseudocomplemented (cf. property (Pe) at the beginning of the previous section) follows from the fact that Σ(P) is lower bounded, that is, the image of a free lattice under a homomorphism η such that for everyS ∈Σ(P),η−1(S) has a least element. The latter property was obtained
in [5] by showing that Σ(P) satisfies a property on join-irreducible elements that was given in [7] as a necessary and sufficient condition for the lower boundedness of afinite lattice. It seems thus illusory to extend this result to the infinite case, but for a finite posetP, it is possible to investigate under what conditions Σ(P) is lower bounded.
The first proof in [13] shows that in a posetPsatisfying the ACC, if we associate to each closure operator ϕthe equivalence relation ≡on P defined by x≡ y iff ϕ(x) = ϕ(y), given a family ϕi (i ∈ I) of closure operators with corresponding equivalence relations ≡i, then the supremum of the ≡i (i ∈ I) in the complete lattice of equivalence relations onP will correspond to the supremum in Φ(P) of theϕi (i∈I). Now this supremum of the ≡i (i∈ I) is the transitive closure of their union, so the result in [13] means that W
i∈Iϕi
(x) = W
i∈Iϕi
(y) iff there is a sequencex=z0, . . . , zn=ysuch that fort= 0, . . . , n−1 there is somei(t)∈I withϕi(t)(zi) =ϕi(t)(zi+1). It can be seen that this holds because for everyx∈P, the set of ϕi1· · ·ϕin(x) (i1, . . . , in ∈I) has a greatest element. However, in case P does not have ACC, this result does not hold anymore (even ifP is a DCPO).
Nevertheless there is a related result in case P is a complete lattice. Given a closure operatorϕonP, since we haveϕ W
i∈Ixi
=ϕ W
i∈Iϕ(xi)
, the relation
≡onP defined byx≡y iff ϕ(x) =ϕ(y), is an equivalence relation closed under supremum: xi ≡yi (i∈ I) implies W
i∈Ixi ≡W
i∈Iyi; such a relation is called a complete join-congruence[3]. Conversely, given a complete join-congruence≡, the operatorϕdefined byϕ(x) =W{y∈P |x≡y}(in fact, the greatest y∈P such that x≡y) is a closure operator, and we obtain in this way a bijection between closure operators on P and complete join-congruences on P. Considering the inclusion order on complete join-congruences, by (1) we obtain that this bijection is an isomorphism between the complete lattice Φ(P) of closure operators onP, and the one of complete join-congruences onP [3]. This result appeared also in [8], where several lattice-theoretical properties of complete join-congruences are presented. The same reference gives also an isomorphism between complete join- congruences and quotient systems: a quotient system is a binary relation ρ on P that is contained in the relation ≤, reflexive, transitive, compatible with the supremum, and such that fora≤c≤d≤b with a ρ b, we must have c ρ d; here the quotient system corresponding to ≡ is given bya ρ b iff a≤ b ≡a, that is, a≤b≤ϕ(a).
In case Σ(P) is closed under arbitrary intersection, it is interesting to consider the decomposition of a closure system as an intersections of some “elementary”
closure systems. Let P be a join-semilattice (i.e., every two elements of P have a join). For any a, b ∈ P, b↑ is a closure system (corresponding to the closure operatorx7→x∨b), andP\a↑ is a down-set; by Corollary 2.2, their union
Fa,b=b↑∪(P\a↑) ={x∈P |a6≤xorb≤x}={x∈P|a≤x ⇒ b≤x} , (5) is a closure system. The closure systemsFa,b (for alla, b∈P) have been studied in depth in the particular case whereP isP(E) for a finite setE [5], where they are calledimplicational closure systems. Given a closure system S corresponding
to a closure operatorϕ, for any a, b ∈ P we have S ⊆ Fa,b ⇔ b ≤ ϕ(a) (i.e., ϕ(b)≤ϕ(a)), while for anyx /∈S we havex /∈Fx,ϕ(x); hence [5]
S =\
{Fa,b|a, b∈P, b≤ϕ(a)}= \
a∈P
Fa,ϕ(a) .
Now if Σ(P) is closed under arbitrary intersection (for example, ifP is a complete lattice), it follows that a subsetS ofP is a closure system iff it is an intersection of implicational closure systemsFa,b. In the case whereP =P(E) for a finite set E, for anyA, B ∈ P(E),FA,B is meet-irreducible iff A⊂S and B is a singleton disjoint fromA[5].
The lattice of closure operators on a complete lattice is isomorphic to that of implication systems[8] (also calledArmstrong systems,FD-systems [8], orfull im- plicational systems[5]. This isomorphism extends to the case of a poset, provided that we modify the definition of an implication system:
LetP be a poset. Animplication systemonP is a binary relationσonP (i.e., a subset ofP2) that:
(1) is transitive: ∀a, b, c∈P, if (a, b),(b, c)∈σ, then (a, c)∈σ;
(2) contains the relation≥: ∀a, b∈P, ifa≥b, then (a, b)∈σ;
(3) is upper bounded on the right: ∀a∈P, the set{b∈ P |(a, b)∈σ} has a greatest element.
Note that ifP is a complete lattice, condition (3) can be replaced by the following, which is the original condition (3) from [8]:
(3’) given{(ai, bi)|i∈I} ⊆σ, we have W
i∈Iai,W
i∈Ibi
∈σ.
Write IS(P) for the poset of implication systems on P (ordered by inclusion).
Then it is easily shown that there is an isomorphism between the two posets Φ(P) andIS(P), where:
• to every closure operatorϕcorresponds the implication systemσϕ={(a, b)∈ P2|b≤ϕ(a)};
• to every implication system σ corresponds the closure operator ϕσ given by setting, for all a ∈ P, ϕσ(a) equal to the greatest element of the set {b∈P |(a, b)∈σ}.
Note that [8] gave also, in case P is a complete lattice, an isomorphism between IS(P) and the complete lattice of of complete join-congruences on P: a ≡ b iff (a, b),(b, a) ∈ σ, and conversely (a, b) ∈ σ iff a ≡ a∨b. Composed with the isomorphism between Φ(P) and IS(P), we obtain the isomorphism given above between closure operators and complete join-congruences, namelya≡biffϕ(a) = ϕ(b).
In [5] this isomorphism between Φ(P) andIS(P) is obtained, forP =P(E), through a Galois connection between P and P2. We show here how it can be
generalized. We define the binary relation∼betweenP2 and P by (a, b)∼f iff a6≤f or b≤f. We obtain thus a Galois connection made of the two maps:
α:P(P2)→ P(P) :σ7→ {f ∈P | ∀(a, b)∈σ, (a, b)∼f} ; β:P(P)→ P(P2) :S7→ {(a, b)∈P2| ∀f ∈S, (a, b)∼f} .
From (5) we see that for any (a, b) ∈ P2, α({(a, b)}) = Fa,b, hence for any σ ∈ P(P2) we have α(σ) = T
(a,b)∈σFa,b. Thus if P is a join-semilattice and Σ(P) is closed under arbitrary intersection (for example, if P is a complete lattice), it follows that Σ(P) = α(P(P2)), the image of α. By the Galois connection, the restriction of β to Σ(P) constitutes a dual isomorphism between Σ(P) and β(P(P)), the image ofβ. Now for a closure systemS corresponding to a closure operatorϕ, (a, b)∈β(S) ⇔ S⊆α({(a, b)}) =Fa,b, hence
β(S) ={(a, b)∈P2|S ⊆Fa,b}={(a, b)∈P2|b≤ϕ(a)}=σϕ , the implication system corresponding toϕ.
Let us conclude. In Section 2 we proved that for a posetP, Σ(P) is detachable and lower semimodular. Then Section 3 showed that many related properties, given in case P is a complete lattice, do not hold in general. In this section we have raised the question of the level of generality of these properties, namely of the minimal conditions on the posetP that guarantee such properties. We have also considered alternative characterizations of closure operators and systems, given in caseP is a complete lattice: complete join-congruences, quotient systems and implication systems. Although the characterization by implication systems extends to the general case of an arbitrary posetP, we do not know if this can also be done for complete join-congruences and quotient systems. Thus further research on the above problems is amply justified.
Acknowledgement. The author is indebted to Marcel Ern´e for pointing out the notion of detachability [15] and its relevance for closure systems [12]. The referee made also many helpful suggestions.
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Received
Christian Ronse, LSIIT UMR 7005 CNRS-UdS, Parc d’Innovation, Boulevard S´ebastien Brant, BP 10413, 67412 ILLKIRCH CEDEX, FRANCE
E-mail: [email protected]
