A poset (partially ordered set) is a pair * (P, ≤)*, where

* P* is a set,

* ≤* is a binary relation on P satisfying the three axioms of partial order:

(i) Reflexive: **∀x ∈ P, x ≤ x**

(ii) Antisymmetric: **∀x ,y ∈ P, x ≤ y & y ≤ x ⇒ x = y**

(iii) Transitive: * ∀x, y, z ∈ P, x ≤ y & y ≤ z ⇒ x ≤ z*.

And what does this have to do with category theory?

**“x ≤ y” ⇐⇒ “ x → y” “x = y” ⇐⇒ “ x ↔ y”**

Given * x, y ∈ P*,

we say that * u ∈ P* is a least upper bound of

*if we have*

**x, y ∈ P***&*

**x → u***, and for all*

**y → u***satisfying*

**z ∈ P***&*

**x → z***we must have*

**y → z***. It is more convenient to express this definition with a picture. We say that*

**u → z***is a least upper bound of*

**u ∈ P***,*

**x***if for all*

**y***the following picture holds:*

**z ∈ P**Dually, we say that * l ∈ P* is a greatest lower bound of

*if for all*

**x, y***the following picture holds:*

**z ∈ P**Now suppose that * u_{1}, u_{2} ∈ P* are two least upper bounds for

*. Applying the defininition in both directions gives*

**x, y*** u_{1} → u_{2}* and

*,*

**u**_{2}→ u_{1}and then from antisymmetry it follows that * u_{1}* =

*, which just means that*

**u**_{2}*and*

**u**_{1}*are indistinguishable within the structure of*

**u**_{2}*. For this reason we can speak of the least upper bound (or “join”) of*

**P***. If it exists, we denote it by*

**x, y****x ∨ y**

Dually, if it exists, we denote the greatest lower bound (or “meet”) by

**x ∧ y**

The definitions of meet and join are called “universal properties”. Whenever an object defined by a universal property exists, it is automatically unique in a certain canonical sense. However, since the object might not exist, maybe it is better to refer to a universal property as a “characterization,” or a “prescription,” rather than a “definition.”

Let * P* be a poset. We say that

*is a top element*

**t ∈ P**if for all * z ∈ P* the following picture holds:

**z —> t**

Dually, we say that * b ∈ P* is a bottom element if for all

*the following picture holds:*

**z ∈ P****b —> z**

For any subset of elements of a poset * S ⊆ P* we say that the element

*is its join if for all*

**⋁ S ∈ P***the following diagram is satisfied:*

**z ∈ P**Dually, we say that * ⋀ S ∈ P* is the meet of

*if for all*

**S***the following diagram is satisfied:*

**z ∈ P**If the objects * ⋁ S* and

*exist then they are uniquely characterized by their universal properties.*

**⋀ S**The universal properties in these diagrams will be called the “limit” and “colimit” properties when we move from posets to categories. Note that a limit/colimit diagram looks like a “cone over S”. This is one example of the link between category theory and topology.

Note that all definitions so far are included in this single (pair of) definition(s):

* ⋁ {x, y} = x∨ y *&

**⋀ {x, y} = x ∧ y*** ⋁∅ = 0* &

*.*

**⋀ ∅ = 1**