Lattices & CPOs — where fixed points live

Parent: order-theory

The completeness ladder

lattice — a poset where every pair has a least upper bound (join ∨) and greatest lower bound (meet ∧).

complete lattice — every subset has a join and a meet (so there's a top ⊤ and bottom ⊥).

CPO / dcpo — a poset with a least element ⊥ and joins of all directed sets (or of all chains).

Why these two exact strengths: they are the homes of the constructive fixed-point theorems (no choice needed):

complete lattice → Knaster–Tarski: a monotone map has a whole complete lattice of fixed points (a least and a greatest).
CPO → Kleene: a Scott-continuous map has a least fixed point ⊔ₙ fⁿ(⊥), reached by climbing from ⊥ — the semantics of recursion.

Examples: the powerset under ⊆ (complete lattice); divisors of n under | (a lattice); partial functions ordered by extension, and "flat" data domains, are CPOs (⊥ = "undefined") — the setting for denotational semantics.