Total order
All
Minimum element
Partial order
Aka Posets: Partially Ordered sets. Maybe some
Minimal element
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
Supremum property in S
S with supremum property: For any
If S has supremum property, it has infemum property: For any subset
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’
Thence get ‘well founded set’. Minimal elements exist.
Eg:
Mathematical induction proofs generalized
(Noether). If