Upper bounds in posets
Content created by Egbert Rijke and Fredrik Bakke.
Created on 2023-05-05.
Last modified on 2024-02-06.
module order-theory.upper-bounds-posets where
Imports
open import foundation.dependent-pair-types open import foundation.propositions open import foundation.universe-levels open import order-theory.posets
Idea
An upper bound of two elements x
and y
in a poset P
is an element z
such that both x ≤ z
and y ≤ z
hold. Similaryly, an upper bound of a
family x : I → P
of elements in P
is an element z
such that x i ≤ z
holds for every i : I
.
Definitions
Binary upper bounds
module _ {l1 l2 : Level} (P : Poset l1 l2) where is-binary-upper-bound-Poset-Prop : (x y z : type-Poset P) → Prop l2 is-binary-upper-bound-Poset-Prop x y z = product-Prop (leq-Poset-Prop P x z) (leq-Poset-Prop P y z) is-binary-upper-bound-Poset : (x y z : type-Poset P) → UU l2 is-binary-upper-bound-Poset x y z = type-Prop (is-binary-upper-bound-Poset-Prop x y z) is-prop-is-binary-upper-bound-Poset : (x y z : type-Poset P) → is-prop (is-binary-upper-bound-Poset x y z) is-prop-is-binary-upper-bound-Poset x y z = is-prop-type-Prop (is-binary-upper-bound-Poset-Prop x y z) module _ {l1 l2 : Level} (P : Poset l1 l2) {a b x : type-Poset P} (H : is-binary-upper-bound-Poset P a b x) where leq-left-is-binary-upper-bound-Poset : leq-Poset P a x leq-left-is-binary-upper-bound-Poset = pr1 H leq-right-is-binary-upper-bound-Poset : leq-Poset P b x leq-right-is-binary-upper-bound-Poset = pr2 H
Upper bounds of families of elements
module _ {l1 l2 : Level} (P : Poset l1 l2) where is-upper-bound-family-of-elements-Poset-Prop : {l : Level} {I : UU l} → (I → type-Poset P) → type-Poset P → Prop (l2 ⊔ l) is-upper-bound-family-of-elements-Poset-Prop {l} {I} f z = Π-Prop I (λ i → leq-Poset-Prop P (f i) z) is-upper-bound-family-of-elements-Poset : {l : Level} {I : UU l} → (I → type-Poset P) → type-Poset P → UU (l2 ⊔ l) is-upper-bound-family-of-elements-Poset f z = type-Prop (is-upper-bound-family-of-elements-Poset-Prop f z) is-prop-is-upper-bound-family-of-elements-Poset : {l : Level} {I : UU l} (f : I → type-Poset P) (z : type-Poset P) → is-prop (is-upper-bound-family-of-elements-Poset f z) is-prop-is-upper-bound-family-of-elements-Poset f z = is-prop-type-Prop (is-upper-bound-family-of-elements-Poset-Prop f z)
Properties
Any element greater than an upper bound of a
and b
is an upper bound of a
and b
module _ {l1 l2 : Level} (P : Poset l1 l2) {a b x : type-Poset P} (H : is-binary-upper-bound-Poset P a b x) where is-binary-upper-bound-leq-Poset : {y : type-Poset P} → leq-Poset P x y → is-binary-upper-bound-Poset P a b y pr1 (is-binary-upper-bound-leq-Poset K) = transitive-leq-Poset P a x _ ( K) ( leq-left-is-binary-upper-bound-Poset P H) pr2 (is-binary-upper-bound-leq-Poset K) = transitive-leq-Poset P b x _ ( K) ( leq-right-is-binary-upper-bound-Poset P H)
Any element greater than an upper bound of a family of elements a
is an upper bound of a
module _ {l1 l2 l3 : Level} (P : Poset l1 l2) {I : UU l3} {a : I → type-Poset P} {x : type-Poset P} (H : is-upper-bound-family-of-elements-Poset P a x) where is-upper-bound-family-of-elements-leq-Poset : {y : type-Poset P} → leq-Poset P x y → is-upper-bound-family-of-elements-Poset P a y is-upper-bound-family-of-elements-leq-Poset K i = transitive-leq-Poset P (a i) x _ K (H i)
Recent changes
- 2024-02-06. Fredrik Bakke. Rename
(co)prod
to(co)product
(#1017). - 2023-05-07. Egbert Rijke. Cleaning up order theory some more (#599).
- 2023-05-05. Egbert Rijke. Cleaning up order theory 3 (#593).
- 2023-05-05. Egbert Rijke. cleaning up order theory (#591).