The poset of closure systems on an infinite poset: Detachability and semimodularity Christian Ronse
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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
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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
Let P be a partially ordered set (from now on poset). Following [2], for any x ∈ P , let us write
x↑ = {y ∈ P | y ≥ x} and x↓ = {y ∈ P | y ≤ x} .
For X ⊆ P , we say that X is a down-set if ∀ x ∈ X, x↓ ⊆ X [2]. Following [14], we call an operator on P a map P → P , and for x ∈ P and an operator ψ on P , the image of x by ψ is written ψ(x); given two operators ξ and ψ on P the operator P → P : x 7→ ψ(ξ(x)) is the composition of ξ followed by ψ, following [14] we write it ψξ rather than the usual ψ ◦ ξ (as in [2]). 2 C. Ronse
A closure operator on P is an operator ϕ that is isotone (x ≤ y ⇒ ϕ(x) ≤ ϕ(y)), extensive (x ≤ ϕ(x)) and idempotent (ϕ(ϕ(x)) = ϕ(x)). Equivalently [9, 20]:
∀ x,y ∈ P, x ≤ ϕ(y) ⇐⇒ ϕ(x) ≤ ϕ(y) .
For any closure operator ϕ, let
Inv(ϕ) = {x ∈ P | ϕ(x) = x} = {ϕ(x) | x ∈ P } be the invariance domain of ϕ [14] (it is also called the range of ϕ [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 on P . It has been known at least since [18] that there is a bijection between Φ(P ) and Σ(P ): to a closure operator ϕ we associate the closure system Inv(ϕ), and conversely to a closure system S we associate the closure operator mapping each x ∈ P to the least element of x↑ ∩ 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 operators ϕ1 and ϕ2 satisfy
ϕ1 ≤ ϕ2 ⇐⇒ Inv(ϕ1) ⊇ Inv(ϕ2) ⇐⇒ ϕ2ϕ1 = ϕ2 ⇐⇒ ϕ1ϕ2 = ϕ2 . (1) Thus the bijection ϕ 7→ 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: for ϕi ∈ Φ(P ) (i ∈ I), i∈I ϕi : x 7→ i∈I ϕi(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 V V intersection: for Si ∈ Σ(P ) (i ∈ I), i∈I Si ∈ Σ(P ) (in particular, for I = ∅, P is the greatest closure system). Given a complete lattice P (withT greatest element 1), {1} is the least element of Σ(P ), and the {1,x} (x ∈ P \{1}) are the atoms of Σ(P ). Thus every S ∈ Σ(P ) is a union of atoms, and the complete lattice Σ(P ) is atomistic. Manara [17] showed that for a complete lattice P , given two closure systems S1 and S2 on P such that S1 ⊂ S2, in the lattice Σ(P ) we have S2 ≻ S1 (i.e., S2 covers S1) 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 . The poset of closure systems 3
(Here ϕ ∨ ϕ′ denotes the join in the lattice Φ(P ), not the elementwise join x 7→ ϕ(x) ∨ ϕ′(x)). As remarked in [4], Ore [21] had shown these two results in the case where the complete lattice P is P(E), the Boolean lattice of all subsets of a set E. 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, iff P is a complete chain. What can we say in case P is not a complete lattice ? Several authors [1, 19, 13] presented proofs that if the poset P satisfies the ACC (ascending chain condition, or equivalently, every directed subset of P has a greatest element), then Σ(P ) is closed under arbitrary intersection (and has P as greatest element), hence it is a complete lattice. This holds in particular if P 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. Let ϕi (i ∈ I) be closure operators with associated closure systems Si; let M be the monoid generated by the ϕi (i ∈ I); then for any x ∈ P , the set of µ(x) for µ ∈ M is directed (for µ, µ′ ∈ M, µµ′ ∈ M and µ(x), µ′(x) ≤ µµ′(x)), hence it has a greatest element ψ(x); it is easily seen that ψ(x) is the least element ↑ of x ∩ i∈I Si, hence i∈I Si is a closure system; then ψ is a closure operator, namely the supremum in Φ(P ) of the ϕi (i ∈ I). TheT second proof inT [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 element 1). A subset S of a poset P is called MLB-closed [22] iff for any X ⊆ S, every maximal lower bound of X must belong to S (in particular for X = ∅, every maximal element of P must belong to S). 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 poset P satisfying the ACC, a subset S of P 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 where P 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 monoid M generated by the closure operators ϕi (i ∈ I), for any x ∈ P , the directed set of all µ(x), for µ ∈ M, has a supremum ψ(x). This gives an extensive and isotone operator ψ, and the least closure operator ϕ ≥ ψ can be con- structed by several methods, see [6], in particular by transfinite induction, as did Inv Inv [16] in the case where P is a complete lattice. Then we have (ϕ) = i∈I (ϕi). Manara’s [17] results about covers and semimodularity extend also to any poset P where Σ(P ) is closed under arbitrary intersection (in particular if PTis a DCPO). 4 C. Ronse
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 containing X, we write it σ(X) and call it the closure system generated by X. In [13] it was shown that if P has ACC, then Σ(P ) satisfies the anti-matroid exchange property: ∀ S ∈ Σ(P ), ∀ x,y ∈ P \ S, x 6= y, y ∈ σ(S ∪ {x}) =⇒ x∈ / σ(S ∪ {y}) . (2) Ern´e[12] showed that for any poset P with Σ(P ) closed under arbitrary intersec- tion, Σ(P ) is detachable: ∀ S ∈ Σ(P ), ∀ x ∈ P \ S, σ(S ∪ {x}) \ {x} ∈ Σ(P ) . (3) This concept comes from [15] (where one says extremally 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 for S1,S2 ∈ Σ(P ) such that S1 ⊂ S2, S2 covers S1 in the lattice Σ(P ) iff S2 \ 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: ∀ S ,S ∈ Σ(P ), S ⊂ S , ∀ x ∈ S \ S , 0 1 0 1 1 0 (4) ∃ S2 ∈ Σ(P ), S0 ∪ {x} ⊆ S2 ⊆ S1, S2 \ {x} ∈ Σ(P ) . 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 join S1 ∨ S2 exists and S1 ∨ S2 ≻ S1, then the meet S1 ∧ S2 exists 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 meet ϕ1 ∧ ϕ2 exists and ϕ1 ≻ ϕ1 ∧ ϕ2, then the join ϕ1 ∨ ϕ2 exists and ϕ1 ∨ ϕ2 ≻ ϕ2.
Here we have no indication on the explicit values of (ϕ1 ∧ ϕ2)(x) and (ϕ1 ∨ ϕ2)(x) for x ∈ 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 where P is an arbitrary infinite poset. However, some of these properties may hold if one makes appropriate assumptions on P . The poset of closure systems 5 2. The main argument
In this section we shall fix a poset P . 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 and S2 ∈ Σ(P ). The closure operators ϕ0, ϕ1, ϕ2 corresponding to S0,S1,S2 satisfy ϕ1 ≤ ϕ2 ≤ ϕ0, and for every x ∈ P we have
ϕ1(x) if ϕ1(x) ∈ A , ϕ2(x) = (ϕ0(x) if ϕ1(x) ∈/ A .
Proof. As S0 ⊆ S1, we get S2 = (S0 \ A) ∪ (S0 ∩ A) ∪ (S1 ∩ A) = S0 ∪ (S1 ∩ A); by the modular equality, we obtain S2 = (S0 ∪ A) ∩ S1, and S0 ⊆ 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 and ϕ1(x) ∈ x ∩ S2, with x ∩ S2 ⊆ x ∩ S1, we deduce that ϕ1(x) ↑ is the least element of x ∩ S2. Set ϕ2(x) = ϕ1(x). ↑ (2) ϕ1(x) ∈/ A. For any y ∈ x ∩ S1, we have ϕ1(x) ≤ y, and as A is 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 .
↑ ↑ Now ϕ0(x) is the least element of x ∩ S0 = x ∩ S2. Set ϕ2(x) = ϕ0(x). ↑ Thus in both cases ϕ2(x) is the least element of x ∩S2, so S2 is a closure system and ϕ2 is a closure operator. We have ϕ2(x) = ϕ1(x) for ϕ1(x) ∈ A, and ϕ2(x) = ϕ0(x) for ϕ1(x) ∈/ A.
An interesting particular case is for S1 = P (that is, ϕ1 is the identity operator): Corollary 2.2. Given a closure system S on P and a down-set A in P , then S ∪ A is a closure system. If ϕ is the closure operator corresponding to S, then the closure operator ϕA corresponding to S ∪ A is given by
x for x ∈ A , ϕA(x) = (ϕ(x) for x∈ / A .
We obtain from Proposition 2.1 the detachability condition (4):
Corollary 2.3. For any S0,S1 ∈ Σ(P ) such that S0 ⊂ S1 and for any x ∈ S1 \S0, there exists S2 ∈ Σ(P ) such that S0 ∪ {x} ⊆ S2 ⊆ S1 and S2 \ {x} ∈ Σ(P ). 6 C. Ronse
↓ ′ ↓ Proof. A = x and A = x \ {x} = {y ∈ P | y This gives thus a generalization of Manara’s first result: Corollary 2.4. For S0,S1 ∈ Σ(P ) such that S0 ⊂ S1, we have S1 ≻ S0 (in Σ(P )) iff S1 \ 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 covers S0 in the smaller poset Σ(P ). It follows by induction that for S0,S1 ∈ Σ(P ) such that S0 ⊂ S1, the interval [S0,S1] in Σ(P ) has finite height iff S1 \ S0 is finite, and then the height of [S0,S1] in Σ(P ) equals the size of S1 \ S0. Returning to Corollary 2.3, writing S3 = 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 ). Given ϕ1, ϕ2 the closure operators corresponding to S1,S2, the closure operator n corresponding to S1 ∩ S2 is (ϕ2ϕ1) ϕ2, where n = |S2 \ S1|. Proof. Let n = |S2 \ S1|. For any x ∈ P , ϕ2(x) ∈ S2. For any y ∈ S2, we have two cases: (1) If y ∈ S1 ∩ S2, then ϕ1(y) = y = ϕ2(y), so ϕ2ϕ1(y) = y. (2) If y ∈ S2 \ S1, then y∈ / Inv(ϕ1), so y < ϕ1(y), and ϕ1(y) ≤ ϕ2ϕ1(y), thus y < ϕ2ϕ1(y). t Let x ∈ P ; for any t ∈ N, (ϕ2ϕ1) ϕ2(x) ∈ S2. If there is some t < n such that t (ϕ2ϕ1) ϕ2(x) ∈ S1 ∩ S2, case 1 yields that t+1 t t (ϕ2ϕ1) ϕ2(x) = ϕ2ϕ1 (ϕ2ϕ1) ϕ2(x) = (ϕ2ϕ1) ϕ2(x) , The poset of closure systems 7 n t and thus (ϕ2ϕ1) ϕ2(x) ∈ S1 ∩ S2. If(ϕ2ϕ1) ϕ2(x) ∈ S2 \ S1 for every t < n, by case 2 we get t n ϕ2(x) < · · · < (ϕ2ϕ1) ϕ2(x) < · · · < (ϕ2ϕ1) ϕ2(x) , n−1 with ϕ2(x),..., (ϕ2ϕ1) ϕ2(x) ∈ S2 \ S1 ; n n as |S2 \ S1| = n, this means that (ϕ2ϕ1) ϕ2(x) ∈/ S2 \ S1, that is (ϕ2ϕ1) ϕ2(x) ∈ n S1 ∩ S2. Thus we have shown that for any x ∈ P , (ϕ2ϕ1) ϕ2(x) ∈ S1 ∩ S2. Now n n for x ∈ S1 ∩ S2, by induction case 1 gives (ϕ2ϕ1) ϕ2(x) = x; hence (ϕ2ϕ1) ϕ2 n is idempotent. Clearly (ϕ2ϕ1) ϕ2 inherits the isotony and extensivity of ϕ1 and n n ϕ2. Therefore (ϕ2ϕ1) ϕ2 is a closure operator, with Inv (ϕ2ϕ1) ϕ2 = S1 ∩ S2, so S ∩ S is a closure system. 1 2 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. For S1,S2 ∈ Σ(P ), if in Σ(P ) the join S1 ∨S2 exists and S1 ∨S2 ≻ S1, then the meet S1 ∧ S2 exists and S2 ≻ S1 ∧ S2. We have then S1 ∨ S2 = S1 ∪ S2 and S1 ∧ S2 = S1 ∩ S2. Proof. Suppose that S1 ∨ S2 exists and S1 ∨ S2 ≻ S1; thus S1 ⊆ S1 ∪ S2 ⊆ S1 ∨ S2. We may not have S1 ∪ S2 = S1, otherwise S2 ⊆ S1 and then S1 ∨ S2 = S1. Hence S1 ⊂ S1 ∪ S2 ⊆ S1 ∨ S2, and by Corollary 2.4, (S1 ∨ S2) \ S1 is a singleton; thus we get S1 ∨ S2 = S1 ∪ S2. Now (S1 ∨ S2) \ S1 = (S1 ∪ S2) \ S1 = S2 \ S1; as this is a singleton, Proposition 2.5 gives S1 ∩ S2 ∈ Σ(P ). Then obviously S1 ∩ S2 is the meet of S1 and S2 in Σ(P ). We have S2 \ (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 meet ϕ1 ∧ ϕ2 exists and ϕ1 ≻ ϕ1 ∧ ϕ2, then the join ϕ1 ∨ ϕ2 exists and ϕ1 ∨ ϕ2 ≻ ϕ2. 3. Some counterexamples to other properties We have shown that the results of Ern´eand Manara, namely that Σ(P ) is detach- able and lower semimodular, can be extended to the case where P is an arbitrary poset. What about the other properties satisfied by Σ(P ) in case P 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 all x ∈ P \ {1}, so it is atomistic. 8 C. Ronse (Pc) If P is a meet-semilattice (i.e., every two elements of P have a meet), then for any S1,S2 ∈ Σ(P ), the join S1 ∨ S2 exists in Σ(P ), it is the set of all x1 ∧ x2 for x1 ∈ S1 and x2 ∈ S2 (NB: elements of S1 ∪ S2 take this form: ∀ x1 ∈ S1, x1 = x1 ∧ ϕ2(x1), where ϕ2(x1) ∈ S2); in particular, if S1 and S2 are finite, then S1 ∨ S2 is finite. (Pd) If P 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 i∈I Si exists in Σ(P ), it is the set of i∈I xi for all choices xi ∈ Si; in particular, Σ(P ) is a complete join-semilattice (i.e., every non-void subset of PW has a supremum). V (Pe) If P = P(E), the Boolean lattice of all subsets of a finite set E, then [5] Σ(P ) is join-semidistributive: ∀ X,Y,Z ∈ Σ(P ), if X ∨ Y = X ∨ Z, then X ∨ Y = X ∨ (Y ∧ Z) ; hence it is join-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 poset P , 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 case P 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 join S1 ∨ 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 of N 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 on N do not necessarily have a meet in Σ(N); for example let S0 = 2N (the set of even naturals) and S1 = 2N+1 (the set of odd naturals), as S0 ∩S1 = ∅, The poset of closure systems 9 there is no closure system contained in both S0 and S1; seen otherwise, S0 and S1 correspond to the two closure operators ϕ0 and ϕ1 that map every natural to the least even (resp., odd) natural above it, and iterating ϕ0ϕ1 on any n ∈ N, the t sequence (ϕ0ϕ1) (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 b c d n 000001 010 011 100 101 110 111 00 01 10 11 (a) 2 (b) 0 1 1 0 ε a b c n a (c) 2 (d) b c 1 0 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 on P can be of two forms: either 10 C. Ronse {a, b, c} ∪ X for any X ⊆ N, or {a, b, c, d} ∪ Y for an infinite 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} and P . However given S0 = {a, b, c, d} ∪ 2N and S1 = {a, b, c, d} ∪ (2N + 1), we have S0,S1 ∈ Σ(P ), with S0 ∩ S1 = {a, b, c, d}, but in Σ(P ) we have S0 ∧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, let S2 = {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 all S ∈ Σ(P ) such that S2 ∨ 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 n ordering, see the Hasse diagram in Figure 1 (b); equivalently, P = n∈N{0, 1} , m n where for x = (x0,...,xm−1) ∈ {0, 1} and y = (y0,...,yn−1) ∈ {0, 1} we have S x ≤ y iff m ≤ n and xi = yi for i = 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 on P is the identity, thus P 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 on N; the Hasse diagram of P is shown in Figure 1 (c). The MLB-closed subsets of P are all those containing a. On the other hand, the closure systems on P are the sets {a} ∪ X, {a, b} ∪ X and {a, c} ∪ X for any X ⊆ 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 have S0,S1 ∈ Σ(P ), with S0 ∩ 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: Let P = A ∪ {a, b, c}, where A is an infinite antichain (∀ x,y ∈ A, x 6 F. Finite/co-finite: Let P be the collection consisting of all finite and co-finite (complement of finite) subsets of N. Then P, 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 family Si ∈ Q (i ∈ I), either i∈I Si ∈ Q or i∈I Si ∈/ P (i.e., i∈I Si is infinite with infinite complement); in other words Q is the trace on P of a closure system (Moore family) on P(E). NowT Σ(P) is notT a lattice, cf. T (F1). Indeed, for any n ∈ N, let An = 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 all An ∪ X for n ∈ 4N (resp., n ∈ 4N + 2) and X a finite subset of 2N + 1; we can check that S0, S1 ∈ Σ(P), but S0 and S1 have no meet in Σ(P), because S0 ∩ S1 is the set of of all co-finite subsets B of N containing 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 poset P 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 of P ? • If P does not have a greatest element, when is Σ(P ) an atomistic complete lattice ? • We showed that for S1,S2 ∈ Σ(P ) such that S2 \S1 is finite, S1 ∩S2 ∈ Σ(P ). What are then the conditions for the join S1 ∨ 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. If P = P(E), the Boolean lattice of all subsets of a finite set E, 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 every S ∈ Σ(P ), η−1(S) has a least element. The latter property was obtained 12 C. Ronse 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 a finite lattice. It seems thus illusory to extend this result to the infinite case, but for a finite poset P , it is possible to investigate under what conditions Σ(P ) is lower bounded. The first proof in [13] shows that in a poset P satisfying the ACC, if we associate to each closure operator ϕ the 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 on P will correspond to the supremum in Φ(P ) of the ϕi (i ∈ I). Now this supremum of the ≡i (i ∈ I) is the transitive closure of their union, so the result in [13] means that i∈I ϕi (x) = i∈I ϕi (y) iff there is a sequence x = z0,...,zn = y such that for t = 0,...,n−1 there is some i(t) ∈ I W W with ϕi(t)(zi) = ϕi(t)(zi+1). It can be seen that this holds because for every x ∈ 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 if P is a DCPO). Nevertheless there is a related result in case P is a complete lattice. Given a closure operator ϕ on P , since we have ϕ i∈I xi = ϕ i∈I ϕ(xi) , the relation ≡ on P defined by x ≡ y iff ϕ(x) = ϕ(y), is an equivalence relation closed under W W supremum: xi ≡ yi (i ∈ I) implies i∈I xi ≡ i∈I yi; such a relation is called a complete join-congruence [3]. Conversely, given a complete join-congruence ≡, the operator ϕ defined by ϕ(x) = {y ∈WP | x ≡ yW} (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 completeW 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 on P , and the one of complete join-congruences on P [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 for a ≤ c ≤ d ≤ b with a ρ b, we must have c ρ d; here the quotient system corresponding to ≡ is given by a 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 operator x 7→ x ∨ b), and P \ a↑ is a down-set; by Corollary 2.2, their union ↑ ↑ Fa,b = b ∪(P \a ) = {x ∈ P | a 6≤ x or b ≤ x} = {x ∈ P | a ≤ x ⇒ b ≤ x} , (5) is a closure system. The closure systems Fa,b (for all a, b ∈ P ) have been studied in depth in the particular case where P is P(E) for a finite set E [5], where they are called implicational closure systems. Given a closure system S corresponding The poset of closure systems 13 to a closure operator ϕ, for any a, b ∈ P we have S ⊆ Fa,b ⇔ b ≤ ϕ(a) (i.e., ϕ(b) ≤ ϕ(a)), while for any x∈ / S we have x∈ / Fx,ϕ(x); hence [5] S = {Fa,b | a, b ∈ P, b ≤ ϕ(a)} = Fa,ϕ(a) . a∈P \ \ Now if Σ(P ) is closed under arbitrary intersection (for example, if P is a complete lattice), it follows that a subset S of P is a closure system iff it is an intersection of implicational closure systems Fa,b. In the case where P = P(E) for a finite set E, for any A, B ∈ P(E), FA,B is meet-irreducible iff A ⊂ S and B is a singleton disjoint from A [5]. The lattice of closure operators on a complete lattice is isomorphic to that of implication systems [8] (also called Armstrong systems, FD-systems [8], or full im- plicational systems [5]. This isomorphism extends to the case of a poset, provided that we modify the definition of an implication system: Let P be a poset. An implication system on P is a binary relation σ on P (i.e., a subset of P 2) 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 , if a ≥ 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 if P 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 i∈I ai, i∈I bi ∈ σ. Write IS(P ) for the poset of implication W systemsW on P (ordered by inclusion). Then it is easily shown that there is an isomorphism between the two posets Φ(P ) and IS(P ), where: • to every closure operator ϕ corresponds the implication system σϕ = {(a, b) ∈ P 2 | 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, namely a ≡ b iff ϕ(a) = ϕ(b). In [5] this isomorphism between Φ(P ) and IS(P ) is obtained, for P = P(E), through a Galois connection between P and P 2. We show here how it can be 14 C. Ronse generalized. We define the binary relation ∼ between P 2 and P by (a, b) ∼ f iff a 6≤ f or b ≤ f. We obtain thus a Galois connection made of the two maps: α : P(P 2) → P(P ) : σ 7→ {f ∈ P | ∀ (a, b) ∈ σ, (a, b) ∼ f} ; β : P(P ) → P(P 2) : S 7→ {(a, b) ∈ P 2 | ∀ f ∈ S, (a, b) ∼ f} . 2 From (5) we see that for any (a, b) ∈ P , α({(a, b)}) = Fa,b, hence for any σ ∈ 2 P(P ) we have α(σ) = (a,b)∈σ Fa,b. Thus if P is a join-semilattice and Σ(P ) is closed under arbitrary intersection (for example, if P is a complete lattice), T it follows that Σ(P ) = α(P(P 2)), 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 of β. Now for a closure system S corresponding to a closure operator ϕ, (a, b) ∈ β(S) ⇔ S ⊆ α({(a, b)}) = Fa,b, hence 2 2 β(S) = {(a, b) ∈ P | S ⊆ Fa,b} = {(a, b) ∈ P | b ≤ ϕ(a)} = σϕ , the implication system corresponding to ϕ. Let us conclude. In Section 2 we proved that for a poset P , 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 poset P that guarantee such properties. We have also considered alternative characterizations of closure operators and systems, given in case P 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 poset P , 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´efor pointing out the notion of detachability [15] and its relevance for closure systems [12]. 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