Theorems · Inductive type · order theory
Order.Ideal
(P : Type u_2) → [LE P] → Type u_2
An ideal on an order P is a subset of P that is
- nonempty
- upward directed (any pair of elements in the ideal has an upper bound in the ideal)
- downward closed (any element less than an element of the ideal is in the ideal).
- Defined in
- Mathlib.Order.Ideal
- Cited by
- 102 results in Mathlib
- Foundations
- Depth 1 from the axioms, rests on 2 definitions · uses no axioms
- Assumes
- LE
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites0
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
Nothing in Mathlib beyond the foundations.
Cited by137
Results whose statement or proof uses this declaration.
- Order.Ideal.IsProperstatement · cited by 23
- Order.Ideal.principalstatement · cited by 12
- Order.Ideal.IsPrimestatement · cited by 11
- Order.Ideal.lowerstatement and proof · cited by 11
- Order.Ideal.toLowerSetstatement and proof · cited by 9
- Order.Ideal.PrimePair.Istatement · cited by 9
- Order.Ideal.IsMaximalstatement · cited by 8
- Order.IsIdeal.toIdealstatement · cited by 6
- Order.PFilter.dualstatement · cited by 5
- Order.Ideal.directedstatement and proof · cited by 5
- Order.Ideal.sup_memstatement and proof · cited by 5
- Order.Ideal.PrimePair.isCompl_I_Fstatement · cited by 5