Naïve set theory MOC

Partially ordered set

A poset or partially ordered set is a set equipped with a Relation set such that (viewed here as a set) is #m/def/order

1. reflexive — for all ,
2. transitive — if and , then
3. antisymmetric — if and , then

So a poset is a Preorder with the additional property of antisymmetry. We may also view Posets as categories. They are themselves objects in . 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 be a poset.

• The maximum or terminal element has
• The minimum or initial element has
• A maximal element has
• A minimal element has
• The least upper bound or join is the smallest such that (hence it is the categorical coproduct)
• The greatest lower bound or meet is the largest such that (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 that is total is called a chain of .

Archetypal examples

Set inclusion

Sets together with form a partially ordered class¹

• reflexive — for any set , .
• transitive — if and , then
• antisymmetric — if and , then .²

Properties

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

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