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 #m/def/order

reflexive — for all $𝑎 \in 𝑆$ , $(𝑎, 𝑎) \in 𝑅$
transitive — if $(𝑎, 𝑏) \in 𝑅$ and $(𝑏, 𝑐) \in 𝑅$ , then $(𝑎, 𝑐) \in 𝑅$
antisymmetric — if $(𝑎, 𝑏) \in 𝑅$ and $(𝑏, 𝑎) \in 𝑅$ , 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 $𝖯 𝗈 𝗌$ . 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 $𝑀 \in 𝑃$ has $𝑝 \in 𝑃 ⟹ 𝑝 \leq 𝑀$
The minimum or initial element $𝑁 \in 𝑃$ has $𝑝 \in 𝑃 ⟹ 𝑁 \leq 𝑝$
A maximal element $𝑚 \in 𝑃$ has $𝑚 \leq 𝑝 ⟹ 𝑚 = 𝑝$
A minimal element $𝑛 \in 𝑃$ has $𝑝 \leq 𝑛 ⟹ 𝑝 = 𝑛$
The least upper bound $𝑎 \land 𝑏$ or join is the smallest $𝑢$ such that $𝑎, 𝑏 \leq 𝑢$ (hence it is the categorical coproduct)
The greatest lower bound $𝑎 \lor 𝑏$ or meet is the largest $ℓ$ such that $ℓ \leq 𝑎, 𝑏$ (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 $\subseteq$ form a partially ordered class¹

reflexive — for any set $𝐴$ , $𝐴 \subseteq 𝐴$ .
transitive — if $𝐴 \subseteq 𝐵$ and $𝐵 \subseteq 𝐶$ , then $𝐴 \subseteq 𝐶$
antisymmetric — if $𝐴 \subseteq 𝐵$ and $𝐵 \subseteq 𝐴$ , then $𝐴 = 𝐵$ .²

Properties

Zorn's lemma, equivalent to the axiom of choice

#state/tidy | #lang/en | #SemBr

I avoid saying poset since considering a set of all sets introduces problems. ↩
This property is often used to prove sets are the same. ↩

Partially ordered set

Further terminology

Archetypal examples

Set inclusion

Properties

Footnotes