# Subsets of abelian groups

Content created by Egbert Rijke, Fredrik Bakke and Maša Žaucer.

Created on 2023-06-08.

module group-theory.subsets-abelian-groups where

Imports
open import foundation.large-locale-of-subtypes
open import foundation.powersets
open import foundation.sets
open import foundation.universe-levels

open import group-theory.abelian-groups
open import group-theory.subsets-groups

open import order-theory.large-locales
open import order-theory.large-posets


## Idea

A subset of an abelian group A is a subtype of the underlying type of A. The large poset of all subsets of A is called the powerset of A.

## Definitions

### The large locale of subsets of an abelian group

module _
{l1 : Level} (G : Ab l1)
where

powerset-large-locale-Ab :
Large-Locale (λ l2 → l1 ⊔ lsuc l2) (λ l2 l3 → l1 ⊔ l2 ⊔ l3) lzero
powerset-large-locale-Ab = powerset-Large-Locale (type-Ab G)


### The large poset of subsets of an abelian group

module _
{l1 : Level} (G : Ab l1)
where

powerset-large-poset-Ab :
Large-Poset (λ l2 → l1 ⊔ lsuc l2) (λ l2 l3 → l1 ⊔ l2 ⊔ l3)
powerset-large-poset-Ab =
large-poset-Large-Locale (powerset-large-locale-Ab G)


### Subsets of abelian groups

subset-Ab :
(l : Level) {l1 : Level} (A : Ab l1) → UU (lsuc l ⊔ l1)
subset-Ab l A = subset-Group l (group-Ab A)

is-set-subset-Ab :
(l : Level) {l1 : Level} (A : Ab l1) → is-set (subset-Ab l A)
is-set-subset-Ab l A = is-set-subset-Group (group-Ab A)