If we consider a subset S ⊆ P, informally speaking, not all elements in S or P\S have the same position. Some elements x are in the “interior” of S, that is ∃y, z ∈ S such that y < x < z; while some others are on its “border”, that is there is no element in S below (or above) them. These elements in the border are called minimal and maximal elements (see Definition 2.8).
Definition 2.8. An element s ∈ S is said to be a minimal (resp. maximal) element in S if all
its strict lower (resp. upper) bounds are outside S. The set of minimal (resp. maximal) elements of S is the set denoted by min(S) (resp. max(S)) and given by:
min(S) = {s ∈ S | ↓ s ∩ S = {s}} max(S) = {s ∈ S | ↑ s ∩ S = {s}}
Note 2.4. One should note that for any S ⊆ P, minimal (maximal) elements are incomparable. In other words, min(S) and max(S) are antichains in (P, ≤).
Example 2.4. Consider Fig. 2.1 (1), let be S = {{a},{a, b},{a, c}} . We have min(S) = {{a}} and max(S) = {{a, b},{a, c}}.
Before going further on characterizing the elements of a subset S, one important observation is made in Lemma 2.1 on the relationship between minimal elements of a subset S and the minimal elements of its up-closure ↑ S.
Lemma 2.1. We have:
min(↑ S) = min(S) max(↓ S) = max(S)
Proof. We prove by double inclusion the property min(↑ S) = min(S):
(⊆) Let x ∈ min(↑ S) and suppose that x 6∈ S. Since x ∈ min(↑ S) we have x ∈↑ S, that is ∃y ∈ S s.t. y ≤ x but y 6= x since x 6∈ S. Thus, ∃y ∈↑ S s.t. y ≤ x but y 6= x. Thus y ∈↓ x∩ ↑ S with y 6= x which contradicts the fact that x ∈ min(↑ S) (i.e. ↓ x∩ ↑ S = {x}). We conclude that x ∈ S. Suppose now that x 6∈ min(S), that ∃y ∈ S s.t. y ≤ x and y 6= x. Hence, y ∈↑ S∩ ↓ x which contradicts the fact that x ∈ min(↑ S). Thus x ∈ min(S). We conclude that min(↑ S) ⊆ min(S).
(⊇) Let x ∈ min(S), thus x ∈↑ S. Suppose that x 6∈ min(↑ S) that is ∃y ∈↑ S such that y ≤ x and y 6= x. Thus ∃z ∈ S such that z ≤ y ≤ x with z 6= x. Hence, z ∈↓ x ∩ S with z 6= x which contradicts the fact that x ∈ min(S). We conclude that x ∈ min(↑ S) or more generally min(S) ⊆ min(↑ S).
One can follow the same steps to show that max(↓ S) = max(S).
We have seen in Example 2.4 that maximal elements of a subset S could be multiple and in such case they are incomparable. On the same example, we have seen that the considered subset S has a unique minimal element which at the same time is below all elements of S, this particular element is said to be a minimum and is formally presented in Definition 2.9.
⊥ a0 a1 .. . > b0 b1
Figure 2.3: there is a breach on the wall for S = {⊥, b0, b1} ∪ {ai| i ∈ N}, i.e. max(S) = {b1}.
Definition 2.9. An element m ∈ S is said to be:
• A minimum or a smallest element of S if it is below all elements of S. Formally: m ∈ S and (∀s ∈ S) m ≤ s
• A maximum or a greatest element of S if it is above all elements of S. Formally: m ∈ S and (∀s ∈ S) s ≤ m
The minimum (resp.maximum) does not necessarily exist. Moreover, if it exists then it is unique. Formally, we say that:
• S has a minimum or S is minimum-handle2if S`∩ S 6= ;. • S has a maximum or S is maximum-handle if Su∩ S 6= ;.
Example 2.5. Consider Fig. 2.1 (1), subset S = {{a},{a, b},{a, c}} is a minimum-handle since {a} is a subset of all elements of S. However, S does not have a maximum since elements {a, b} and {a, c} are incomparable and has no element above them in S (i.e. they are maximal elements).
Note 2.5. If the poset has a maximum, we call it the top element and we denote it >. Dually,
if the poset has a minimum, we call it the bottom element and we denote it ⊥.
It is important to understand the difference between maximal elements and the maximum element. In fact, when a subset S has a maximum m then max(S) = {m}. In other words, “If S has a maximum then S has a unique maximal element”. However, the converse (i.e. “If S has a unique maximal element then S has a maximum”) is not true. In fact, this statement holds for finite posets but does not necessarily hold for an arbitrary poset. Consider, for instance the infinite poset (P, ≤) depicted in Fig. 2.3 where P = {⊥,>, b0, b1} ∪ {ai| i ∈ N} with:
• ⊥ ≤ b0≤ b1≤ >.
• (∀i ∈ N) ⊥ ≤ ai, ai≤ ai+1and ai≤ >.
2The terms minimum-handle and maximum-handle used in Definition 2.9 and the terms minimal-handle and maximal-handle used in Definition 2.10 come from Martinez et al’s. [129] paper on multilattices.
Consider, the subset S = P\{>} = {⊥, b0, b1} ∪ {ai| i ∈ N}. It is clear that: max(S) = {b1}. That is,
S has a single maximal element. Yet, S has no maximum (i.e. is not a maximum-handle) since b1is incomparable with elements aifor all i ∈ N. In fact, elements ai has no maximal elements
in max(S) above them (i.e. S 6⊆↓ max(S)). Definition 2.10 formalizes the following (intuitive) property (i.e. every element in S has at least one maximal element above it).
Definition 2.10. We say that S is:
• A minimal-handle if ∀s ∈ S, ∃m ∈ min(S) such that m ≤ s. In other words: S ⊆↑ min(S)
• A maximal-handle if ∀s ∈ S, ∃m ∈ max(S) such that s ≤ m. In other words: S ⊆↓ max(S)
Note that for an upper ideal S ∈U(S) (i.e. S =↑ S), saying that S is minimal-handle is equivalent to say that S =↑ min(S). One should note also that in the case where S is minimal- handle and S has a unique minimal element then S is minimum-handle. It is also worthwhile to notice that, trivially, for any poset we have that ; is a minimal-handle and a maximal-handle since ↑ ; =↓ ; = ;. However, it is neither minimum-handle nor maximum-handle since the ; is empty by its essence and does not contain any element.
Definition 2.11. We have:
• The largest lower bound of S (i.e. the maximum of S`) if it exists is called the infimum or the meet of S and is denoted in f (S) orV S. Moreover, we have S`=↓ (V S), that is:
(∀p ∈ P) p ∈ S`⇐⇒ p ≤^ S
• The smallest upper bound of S (i.e. the minimum of Su) if it exists is called the supremum or the join of S and is denoted su p(S) orW S. Moreover, we have Su=↑ (W S), that is:
(∀p ∈ P) p ∈ Su⇐⇒_ S ≤ p
There is a tight relationship between the minimum and the infimum. In fact, it is easy to see that if S has a minimum then the infimum of S is its minimum. In other words, having the minimum is a stronger property than having an infimum. One important remark is the fact that for any S ⊆ P, Suhas an infimum if and only if it has a minimum. That is, having a minimum is no longer a stronger property than having an infimum when a set of upper bounds is considered. Lemma 2.2 formalizes and generalizes this observation and its dual.
Lemma 2.2. Let (P, ≤) be a poset, S ⊆ P, we have:
• For any A ⊆ S`, if A has a joinW A ∈ P then W A ∈ S`. • For any A ⊆ Su, if A has a meetV A ∈ P then V A ∈ Su.
Proof. Let A ⊆ S`, we have by definition: (∀s ∈ S ∀a ∈ A) a ≤ s, that is S ⊆ Au. SinceW A is the least upper bound of A and all elements of S are upper bounds of A then: (∀s ∈ S)W A ≤ s. We conclude that W A ∈ S`. Same steps can be followed to show the second part of the Lemma.
Proposition 2.2. In case of existence, we have:
^ S = _ S` _ S = ^ Su
Proof. Suppose that S has its infimumV S. By definition, V S is the maximum of S`. Let us show now that this maximum is the join of S`that is the minimum of¡S`¢u
. SinceV S is the maximum of S` thenV S is an upper bound of S` (i.e.V S ∈ V¡S`¢u
). Moreover, let u ∈¡S`¢u
, then (∀x ∈ S`) x ≤ u. Hence, since V S ∈ S`, we have for all u ∈¡S`¢u
: V S ≤ u. SinceV S ∈ V¡S`¢u
thenV S is the minimum of¡S`¢u
(i.e.V S = W Su). One can prove the other property dually.
Note 2.6. According to Proposition 2.2 and since for a poset (P, ≤) we have ;`= ;u= P, then the empty set has a meet (resp. join) iff the poset is upper-bounded (resp. lower bounded):
^ ; = _ P = > _ ; = ^ P = ⊥.