Lower bounds in posets
Content created by Egbert Rijke and Fredrik Bakke.
Created on 2023-05-07.
Last modified on 2024-02-06.
module order-theory.lower-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
A lower bound of two elements a
and b
in a poset P
is an element x
such that both x ≤ a
and x ≤ b
hold. Similarly, a lower bound of a
family a : I → P
of elements in P
is an element x
such that x ≤ a i
holds for every i : I
.
Definitions
Binary lower bounds
module _ {l1 l2 : Level} (P : Poset l1 l2) (a b x : type-Poset P) where is-binary-lower-bound-Poset-Prop : Prop l2 is-binary-lower-bound-Poset-Prop = product-Prop (leq-Poset-Prop P x a) (leq-Poset-Prop P x b) is-binary-lower-bound-Poset : UU l2 is-binary-lower-bound-Poset = type-Prop is-binary-lower-bound-Poset-Prop is-prop-is-binary-lower-bound-Poset : is-prop is-binary-lower-bound-Poset is-prop-is-binary-lower-bound-Poset = is-prop-type-Prop is-binary-lower-bound-Poset-Prop module _ {l1 l2 : Level} (P : Poset l1 l2) {a b x : type-Poset P} (H : is-binary-lower-bound-Poset P a b x) where leq-left-is-binary-lower-bound-Poset : leq-Poset P x a leq-left-is-binary-lower-bound-Poset = pr1 H leq-right-is-binary-lower-bound-Poset : leq-Poset P x b leq-right-is-binary-lower-bound-Poset = pr2 H
Lower bounds of families of elements
module _ {l1 l2 l3 : Level} (P : Poset l1 l2) {I : UU l3} (x : I → type-Poset P) where is-lower-bound-family-of-elements-Poset-Prop : type-Poset P → Prop (l2 ⊔ l3) is-lower-bound-family-of-elements-Poset-Prop z = Π-Prop I (λ i → leq-Poset-Prop P z (x i)) is-lower-bound-family-of-elements-Poset : type-Poset P → UU (l2 ⊔ l3) is-lower-bound-family-of-elements-Poset z = type-Prop (is-lower-bound-family-of-elements-Poset-Prop z) is-prop-is-lower-bound-family-of-elements-Poset : (z : type-Poset P) → is-prop (is-lower-bound-family-of-elements-Poset z) is-prop-is-lower-bound-family-of-elements-Poset z = is-prop-type-Prop (is-lower-bound-family-of-elements-Poset-Prop z)
Properties
Any element less than a lower bound of a
and b
is a lower bound of a
and b
module _ {l1 l2 : Level} (P : Poset l1 l2) {a b x : type-Poset P} (H : is-binary-lower-bound-Poset P a b x) where is-binary-lower-bound-leq-Poset : {y : type-Poset P} → leq-Poset P y x → is-binary-lower-bound-Poset P a b y pr1 (is-binary-lower-bound-leq-Poset K) = transitive-leq-Poset P _ x a ( leq-left-is-binary-lower-bound-Poset P H) ( K) pr2 (is-binary-lower-bound-leq-Poset K) = transitive-leq-Poset P _ x b ( leq-right-is-binary-lower-bound-Poset P H) ( K)
Any element less than a lower bound of a family of elements a
is a lower 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-lower-bound-family-of-elements-Poset P a x) where is-lower-bound-family-of-elements-leq-Poset : {y : type-Poset P} → leq-Poset P y x → is-lower-bound-family-of-elements-Poset P a y is-lower-bound-family-of-elements-leq-Poset K i = transitive-leq-Poset P _ x (a i) (H i) K
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).