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Orders over set

Total order

All \((x, y) \in S\) comparable.

Minimum element

\(\forall y \in S: x\leq y\).

Partial order

Aka Posets: Partially Ordered sets. Maybe some \((x, y) \in S\) not comparable. Eg: vectors, componentwise \(<\). Visualize with Hasse diagrams.

Minimal element

\(\forall y \in S: x\leq y \implies x = y\). Note difference from minimum. Eg: in triangle joining (1,1), (0, 1), (1, 0): line joining the last 2 pts are minimal elements.

Bounds on A, subset of S

Upper and lower bounds: need not be in A.

Supremum or least upper bound or GLB or join. Infimum or greatest lower bound or LUB or meet.

Difference from max and min

\(\max(A) \in A \subseteq S\) always, but \(\sup(A) \notin A, \sup(A) \in S\) possible.

Supremum property in S

S with supremum property: For any \(A \subseteq S: \exists \sup A \in S\).

If S has supremum property, it has infemum property: For any subset \(A \neq S\), take S-A and find its supremum.

Examples

Q does not have supremum property, but R does. See complex analysis survey.

Lattices

Sets where every pair has a same supremum and same infemum: Diamond.

Well founded order

Take any ‘decreasing chain’ \(a<b<c..\): it must be finite. So, \(a<a\) not allowed, no cycles too. Not total or partial order: no transitivity etc required.

Thence get ‘well founded set’. Minimal elements exist.

Eg: \((N, <)\), (strings, psurveyix), (strings, subsequence), (trees, subtree). Lexicographic ordering of \(((a, b), <)\) is well founded if \(<\) well founded over dom(a) and dom(b).

Mathematical induction proofs generalized

(Noether). If \(\forall y:: x>y \land p(y) \implies p(x)\), then \(\forall x: p(x)\); note: base cases subsumed here. Strictly more powerful than induction on natural numbers: consider lexicographic ordering.