Order theory — the zoo of orders
Parent: math
A binary relation ≤ on a set, classified by which axioms it satisfies — reflexive (a≤a), transitive (a≤b≤c ⟹ a≤c), antisymmetric (a≤b ∧ b≤a ⟹ a=b), total (a≤b or b≤a). Each cluster gets a child.
Children
- basic-orders — preorder / partial / total (linear) / strict; chains, antichains, Hasse diagrams.
- well-foundedness — well-founded relations & well-orders; ordinals; well-founded / transfinite induction.
- lattices-and-cpos — lattice / complete lattice / CPO — where fixed points live (Knaster–Tarski, Kleene).
- zorns-lemma — chains-have-upper-bounds ⟹ a maximal element.
- choice-zorn-wellordering — the trio: AC ⟺ Zorn ⟺ well-ordering (equivalent over ZF).
- equivalence-relation — reflexive + symmetric + transitive; the order family's twin.
Where each shows up in the vault
well-founded → [[pc-well-founded-recursion|P×C-well-founded recursion]]; well-order/ordinals → proof-theoretic ordinals; CPO/complete lattice → kleene-fixed-point; chains+upper bounds → zorns-lemma.