# Meet-suplattices

Content created by Egbert Rijke, Fredrik Bakke, Julian KG, fernabnor and louismntnu.

Created on 2023-05-08.

module order-theory.meet-suplattices where

Imports
open import foundation.binary-relations
open import foundation.dependent-pair-types
open import foundation.propositions
open import foundation.sets
open import foundation.universe-levels

open import order-theory.meet-semilattices
open import order-theory.posets
open import order-theory.suplattices


## Idea

An l-meet-suplattice is a meet-semilattice L which has least upper bounds for all families of elements x : I → L indexed by a type I : UU l.

Note that meet-suplattices are not required to satisfy a distributive law. Such meet-suplattices are called frames.

## Definitions

### The predicate on meet-semilattices of being a meet-suplattice

module _
{l1 : Level} (l2 : Level) (X : Meet-Semilattice l1)
where

is-meet-suplattice-Meet-Semilattice-Prop : Prop (l1 ⊔ lsuc l2)
is-meet-suplattice-Meet-Semilattice-Prop =
is-suplattice-Poset-Prop l2 (poset-Meet-Semilattice X)

is-meet-suplattice-Meet-Semilattice : UU (l1 ⊔ lsuc l2)
is-meet-suplattice-Meet-Semilattice =
type-Prop is-meet-suplattice-Meet-Semilattice-Prop

is-prop-is-meet-suplattice-Meet-Semilattice :
is-prop is-meet-suplattice-Meet-Semilattice
is-prop-is-meet-suplattice-Meet-Semilattice =
is-prop-type-Prop is-meet-suplattice-Meet-Semilattice-Prop


### Meet-suplattices

Meet-Suplattice : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)
Meet-Suplattice l1 l2 =
Σ (Meet-Semilattice l1) (is-meet-suplattice-Meet-Semilattice l2)

module _
{l1 l2 : Level} (A : Meet-Suplattice l1 l2)
where

meet-semilattice-Meet-Suplattice : Meet-Semilattice l1
meet-semilattice-Meet-Suplattice = pr1 A

poset-Meet-Suplattice : Poset l1 l1
poset-Meet-Suplattice =
poset-Meet-Semilattice meet-semilattice-Meet-Suplattice

type-Meet-Suplattice : UU l1
type-Meet-Suplattice =
type-Poset poset-Meet-Suplattice

leq-meet-suplattice-Prop : (x y : type-Meet-Suplattice) → Prop l1
leq-meet-suplattice-Prop = leq-Poset-Prop poset-Meet-Suplattice

leq-Meet-Suplattice : (x y : type-Meet-Suplattice) → UU l1
leq-Meet-Suplattice = leq-Poset poset-Meet-Suplattice

is-prop-leq-Meet-Suplattice :
(x y : type-Meet-Suplattice) → is-prop (leq-Meet-Suplattice x y)
is-prop-leq-Meet-Suplattice = is-prop-leq-Poset poset-Meet-Suplattice

refl-leq-Meet-Suplattice : is-reflexive leq-Meet-Suplattice
refl-leq-Meet-Suplattice = refl-leq-Poset poset-Meet-Suplattice

antisymmetric-leq-Meet-Suplattice : is-antisymmetric leq-Meet-Suplattice
antisymmetric-leq-Meet-Suplattice =
antisymmetric-leq-Poset poset-Meet-Suplattice

transitive-leq-Meet-Suplattice : is-transitive leq-Meet-Suplattice
transitive-leq-Meet-Suplattice = transitive-leq-Poset poset-Meet-Suplattice

is-set-type-Meet-Suplattice : is-set type-Meet-Suplattice
is-set-type-Meet-Suplattice = is-set-type-Poset poset-Meet-Suplattice

set-Meet-Suplattice : Set l1
set-Meet-Suplattice = set-Poset poset-Meet-Suplattice

is-suplattice-Meet-Suplattice :
is-suplattice-Poset l2 poset-Meet-Suplattice
is-suplattice-Meet-Suplattice = pr2 A

suplattice-Meet-Suplattice : Suplattice l1 l1 l2
suplattice-Meet-Suplattice =
( poset-Meet-Suplattice , is-suplattice-Meet-Suplattice)

meet-Meet-Suplattice :
(x y : type-Meet-Suplattice) →
type-Meet-Suplattice
meet-Meet-Suplattice =
meet-Meet-Semilattice meet-semilattice-Meet-Suplattice

sup-Meet-Suplattice :
{I : UU l2} → (I → type-Meet-Suplattice) →
type-Meet-Suplattice
sup-Meet-Suplattice {I} f = pr1 (is-suplattice-Meet-Suplattice I f)