Subsets of groups

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

Created on 2023-06-08.
Last modified on 2024-04-20.

module group-theory.subsets-groups where
Imports 
open import foundation.large-locale-of-subtypes
open import foundation.sets
open import foundation.universe-levels

open import group-theory.groups

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

Idea

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

Definitions

The large locale of subsets of a group

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

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

The large poset of subsets of a group

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

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

Subsets of groups

subset-Group :
  (l : Level) {l1 : Level} (G : Group l1) → UU (lsuc l ⊔ l1)
subset-Group l G = type-Large-Locale (powerset-large-locale-Group G) l

is-set-subset-Group :
  {l1 l2 : Level} (G : Group l1) → is-set (subset-Group l2 G)
is-set-subset-Group G =
  is-set-type-Large-Locale (powerset-large-locale-Group G)

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