2.3 Partially Ordered Sets Properties
2.3.5 Multilattices and Complete Multilattices
The term multilattice was introduced for the first time by Benado in [22]. This notion has not received much interest for a long period, but has been unearthed and revisited in the beginning of the 21stcentury [50, 128, 129] for other purpose. While this may seem illogical, we will start by presenting multilattices following [50, 128, 129]. We will then understand the main difference between Benado’s multilattices [22] and Martinez’s multilattices [50, 128, 129] afterward.
Multilattices, as their names imply, are related in their definition with lattices. Simply put, multilattices are a relaxation of lattices where rather than demanding that the set of lower (resp. upper) bounds of each nonempty finite subset is minimum-handle (resp. maximum-handle), multilattices demand that the set of lower (resp. upper) bounds of each nonempty finite subset is minimal-handle (resp. maximal-handle).
Definition 2.18. A poset (P, ≤) is said to be:
• A meet-multisemilattice if for all nonempty finite S ⊆ P we have S`is maximal-handle, that is:
S`= ↓ max³S`´ (M)
The set max(S`) is called the multi-infimum of S and is denoted minf(S). • A complete meet-multisemilattice if condition (M) holds for all S ⊆ P.
• A join-multisemilattice if for all non empty subsets S ⊆ P we have Suis minimal-handle, that is:
Su= ↑ min¡Su¢ (J)
The set min(Su) is called the multi-supremum of S and is denoted msup(S). • A complete join-multisemilattice if condition (J) holds for all S ⊆ P.
Note that when property (M) (resp. property (J)) holds for some subset S ⊆ P, we will say that S has all its multi-infima (resp. S has all its multi-suprema).
abc0 abc1 .. . abcn .. . a b c ab0 ab1 . . . abn . . . ac0 ac1 . . . acn . . . bcn . . . bc1 bc0
Figure 2.5: For all x, y in this poset, {x, y} has all its multi-infima. That is, this poset is a multistructure following [22], however it is not a multilattice following definition 2.18.
Note 2.13. It is clear that all finite posets, or more generally chain-finite posets, are complete multilattices. Hence, conversely to lattices, the notion of multilattices has no importance when finite posets are considered. One should also note that all lattices are multilattices and all complete lattices are complete multilattices.
Example 2.13. Rather than giving an example about a multilattice, we show here that the property of being a multilattice does not trivially hold. Consider for instance the poset (P, ≤) depicted in Fig. 2.6 where P = {ci| i ∈ N} ∪ {a, b} s.t. (∀i ∈ N) ci≤ ci+1, ci≤ a and ci≤ b. It is clear
that {a, b}`= {ci| i ∈ N}. Moreover, max({ci| i ∈ N}) = ;. Hence, {a, b}`6=↓ max({a, b}`). Therefore, (P, ≤) is not a meet-multisemilattice.
In the following of this section, we revisit some important differences between multilattices and lattices. We will try to answer the three properties discussed on lattices:
1. Are pairs the building blocks of a multilattice?
2. Are complete meet-multisemilattice complete multilattices?
3. What is the relationship between complete multilattices and chain-completeness?
2.3.5.1 The pairs are no longer the building blocks
Recall that the pairs are the building blocks of lattices as we have seen in section 2.3.4.1. But does this property remain for multilattices? Let us start by defining the two following conditions for a given poset (P, ≤):
(∀x, y ∈ P) {x, y}`=↓ max³{x, y}`´ (MM2)
(∀x, y ∈ P) {x, y}u=↑ min¡{x, y}u¢ (MJ2)
In fact, Benado defined multilattices in the seminal paper [22] as posets for which condition (MM2) and condition (MJ2) hold. Such posets will be called Benado’s multilattices in this paper.
It is clear that all multilattices following definition 2.18 are Benado’s multilattices. However, do we have an equivalence? The answer is negative as shown in [129] and in the following counter-example.
Example 2.14. Let be the poset (P, ≤) depicted in Fig. 2.5 where P = {abci| i ∈ N} ∪ {abi| i ∈
N} ∪ {aci| i ∈ N} ∪ {bci| i ∈ N} ∪ {a, b, c} and:
• (∀i ∈ N) abci≤ abi, abci≤ aciand abci≤ bci.
• (∀i ∈ N) abi≤ a and abi≤ b.
• (∀i ∈ N) aci≤ a and aci≤ c.
• (∀i ∈ N) bci≤ b and bci≤ c.
One could verify that this poset is a Benado’s multilattice (i.e. both (MM2) and (MJ2) conditions hold), however, this poset is still not a multilattice following definition 2.18. Indeed, considering the non empty finite set {a, b, c}, we have {a, b, c}`= {abci| i ∈ N} while max({a, b, c}`) = ;. It
follows that {a, b, c}`6=↓ max({a, b, c}`).
Hence, the pairs are no longer the building blocks of a multilattice (i.e. being a multilattice following Definition 2.18 is a more restrictive property than being a Benado’s multilattice).
2.3.5.2 No more “Buy one, get one for free”
Again, another adage for lattices (P, ≤) is the “Buy one, get one for free”. Indeed, if all subsets in P have their meet then all subsets have also their joins. In other words, complete semilattices are complete lattices. However, this adage does no longer hold for multilattices. In fact, a complete join-multisemilattice could be not a meet-multisemilattice. Indeed, Fig. 2.6 presents a complete join-multisemilattice since it has the Descending Chain Condition. Yet, it is not a meet-multisemilattice (i.e. {a, b}`= {ci| i ∈ N} with max({a, b}`) = ;). It follows that the “buy one,
get one for free” adage does no longer hold for complete multilattices.
2.3.5.3 Chain-complete posets are complete meet-multisemilattice
We have seen that all chain-complete lattices are complete lattices and vice-versa. What is then the relationship between complete multilattices and chain-complete posets? Theorem 2.5 gives an answer to this question. However, the proof of this theorem requires the usage of the so called Zorn’s Lemma4.
Definition 2.19 (Zorn’s Lemma). Let (P, ≤) be a poset, if every chain in (P,≤) has an upper-
bound, then (P, ≤) has a maximal element. Formally:
(∀C ∈C(P)) Cu6= ; =⇒ max(P) 6= ;
A stronger statement, yet equivalent, of Zorn’s Lemma is stated in the following theorem: 4I am grateful to JOZEFPÓCSfor attracting my attention to Zorn’s Lemma.
a b .. . cn .. . c1 c0
Figure 2.6: A complete join-
multisemilattice but not a meet- multisemilattice c0 c1 .. . cn .. . e0 e1 a0 a1 . . . an . . .
Figure 2.7: A complete multilattice that is not chain-complete
Theorem 2.4 (Zorn’s Lemma II). Let (P, ≤) be a poset, we have:
(∀C ∈C(P)) Cu6= ; =⇒ P =↓ max(P)
Zorn’s Lemma need to be considered as an axiom since it is equivalent to axiom of choice (AC) (see Definition 2.1).
Theorem 2.5. Under Axiom of Choice (AC) assumption, we have: • All chain-complete posets are complete meet-multisemilattice. • All dually chain-complete posets are complete join-multisemilattice. • All doubly chain-complete posets are complete multilattices.
Proof. We show here the first statement of the theorem. Let (P, ≤) be a chain-complete poset and let S ⊆ P. We show here that S`=↓ max(S`). It is straightforward by definition and independently from any assumption that ↓ max(S`) ⊆ S`. It remains to show that S`⊆↓ max(S`). Since (P, ≤) is chain-complete, then every C ⊆ S`has its joinW C ∈ P. Hence, according to Lemma 2.2 and since C ⊆ S` thenW C ∈ S`. Thus every chain C in the sub-poset (S`, ≤) has an upper boundW C ∈ S`. According to Zorn’s Lemma, and by recalling that the Axiom of choice is equivalent to Zorn’s Lemma , we have S`=↓ max(S`). Hence, (P, ≤) is a complete meet-multisemilattice. The other statements can be showed dually.
Note 2.14. Please note that double chain-completeness is only a sufficient condition (under the Axiom of Choice) to have a complete multilattice but not a necessary one. Indeed, one can show that the poset depicted in Fig. 2.7 is a complete multilattice (Remark that ∀i ∈ N : ci≤ ai,
ci≤ ci+1, ai≤ e0and ai≤ e1) but not chain-complete since the chain C = {ci| i ∈ N} does not have a join. Indeed, Cu= {e0, e1}which is an antichain (i.e. Cu has two minimal elements).
2.3.6 Summary
Table 2.2 summarizes the different poset properties that we have learned so far. Please refer to Table 2.1 of the different notations used here. Fig. 2.8 depicts different posets properties and their relationship. One should note that some links are broken because of the empty set (i.e. the non existence of the top and/or the bottom element). For instance, bounded chain-finite posets are doubly chain-complete.
Last but not least, we invite the reader attention to the notion of the direct product between posets that we will use in this manuscript.
Definition 2.20. The direct product of the two posets (P1, ≤) and (P2, ≤) is the poset denoted
(P1, ≤) × (P2, ≤) and given by:
(P1, ≤) × (P2, ≤) := (P1× P2, ≤)
(x1, x2) ≤ (y1, y2) ⇐⇒ x1≤ y1and x2≤ y2
More generally, let I be an arbitrary Index set and let (Pi, ≤) be a poset for each i ∈ I. The
direct product of posets (Pi, ≤) is the poset denoted by
×
i∈I(Pi, ≤) and given by:×
i∈I(Pi, ≤) :=µ
×
i∈IPi, ≤¶
(xi)i∈I≤ (yi)i∈I ⇐⇒ (∀i ∈ I) xi≤ yi
Interestingly, the direct product poset preserves all its properties from the starting posets. For instance, if for all i ∈ I, poset (Pi, ≤) are complete lattices then the direct-product poset is also
a complete lattice. Moreover: (∀S ⊆
×
i∈I
Pi) ^ S = ¡^ Si¢i∈I and _ S = ¡_ Si¢i∈I with Si=©xi| (xj)j∈I∈ Sª
Properties of being bounded, chain-finite, chain-complete or complete multilattices are also transferred to the direct product poset.
Property Meaning with (P, ≤) poset Finite P is finite
ACC or maximal condition (∀S ∈℘(P)) S is maximal-handle (i.e. S ⊆↓ max(C)) DCC or minimal condition (∀S ∈℘(P)) S is minimal-handle (i.e. S ⊆↑ min(C))
Chain-finite Both properties above (i.e. all chains are finite) Chain-complete (∀C ∈C(P)) C has a supremum
Dually Chain-complete (∀C ∈C(P)) C has an infimum Doubly Chain-complete Both properties above
Upper-bounded Set P has a maximum called the top and denoted > Lower-bounded Set P has a minimum called the bottom and denoted ⊥
Bounded Both properties above
Complete Meet-multisemilattice (∀S ∈℘(P)) S` is maximal-handle Complete Join-multisemilattice (∀S ∈℘(P)) Suis minimal-handle
Complete Multilattice Both properties above
Meet-multisemilattice (∀S ∈℘(P) | S nonempty and finite) S` is maximal-handle Join-multisemilattice (∀S ∈℘(P) | S nonempty and finite) Suis minimal-handle
Multilattice Both properties above
Benado’s Multilattice (∀p, q ∈ P) {p, q}` is maximal-handle and {p, q}uis minimal-handle Complete lattice (∀S ∈℘(P)) S` is maximum-handle and
Suis minimum-handle Meet-semilattice (∀p, q ∈ P) {p, q}` is maximum-handle
Join-semilattice (∀p, q ∈ P) {p, q}uis minimum-handle Lattice Both properties above
finite chain-finite antichain-finite ACC doubly chain-complete chain-complete complete multilattice complete meet-multisemilattice meet-multisemilattice multilattice Benado’s multilattice complete lattice lattice meet-semilattice
Under (AC) assumption Implication
Figure 2.8: Posets properties and their relationships. The dual properties (DCC, join- multisemilattices, etc.) and their relationships can be deduced analgeously.