Naïve set theory MOC

Partially ordered set

A poset or partially ordered set is a set 𝑆 equipped with a Relation set 𝑅 such that 𝑅 (viewed here as a set) is order

1. reflexive — for all 𝑎 ∈𝑆, (𝑎,𝑎) ∈𝑅
2. transitive — if (𝑎,𝑏) ∈𝑅 and (𝑏,𝑐) ∈𝑅, then (𝑎,𝑐) ∈𝑅
3. antisymmetric — if (𝑎,𝑏) ∈𝑅 and (𝑏,𝑎) ∈𝑅, then 𝑎 =𝑏

So a poset is a Preorder with the additional property of antisymmetry. We may also view Posets as categories. They are themselves objects in Category of posets. If the poset has the additional property of being total, i.e. all elements are related in some way, it is a Totally ordered set.

Further terminology

Let 𝑃 be a poset.

• The maximum or terminal element 𝑀 ∈𝑃 has 𝑝 ∈𝑃 ⟹ 𝑝 ≤𝑀
• The minimum or initial element 𝑁 ∈𝑃 has 𝑝 ∈𝑃 ⟹ 𝑁 ≤𝑝
• A maximal element 𝑚 ∈𝑃 has 𝑚 ≤𝑝 ⟹ 𝑚 =𝑝
• A minimal element 𝑛 ∈𝑃 has 𝑝 ≤𝑛 ⟹ 𝑝 =𝑛
• The least upper bound sup{𝑎,𝑏} or join is the smallest 𝑢 such that 𝑎,𝑏 ≤𝑢 (hence it is the categorical coproduct)
• The greatest lower bound inf{𝑎,𝑏} or meet is the largest ℓ such that ℓ ≤𝑎,𝑏 (hence it is the categorical product)
• A poset for which every pair of elements have a l.u.b. and g.l.b. is called a Lattice order.
• A subset of 𝑃 that is total is called a chain of 𝑃.

Archetypal examples

Set inclusion

Sets together with ⊆ form a partially ordered class¹

• reflexive — for any set 𝐴, 𝐴 ⊆𝐴.
• transitive — if 𝐴 ⊆𝐵 and 𝐵 ⊆𝐶, then 𝐴 ⊆𝐶
• antisymmetric — if 𝐴 ⊆𝐵 and 𝐵 ⊆𝐴, then 𝐴 =𝐵.²

Properties

• Zorn’s lemma, equivalent to the axiom of choice

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Footnotes

1. I avoid saying poset since considering a set of all sets introduces problems. ↩

2. This property is often used to prove sets are the same. ↩