Subsets of semirings

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

Created on 2023-03-18.

module ring-theory.subsets-semirings where

Imports
open import foundation.identity-types
open import foundation.propositional-extensionality
open import foundation.propositions
open import foundation.sets
open import foundation.subtypes
open import foundation.universe-levels

open import ring-theory.semirings


Idea

A subset of a semiring is a subtype of the underlying type of a semiring

Definition

Subsets of semirings

subset-Semiring :
(l : Level) {l1 : Level} (R : Semiring l1) → UU (lsuc l ⊔ l1)
subset-Semiring l R = subtype l (type-Semiring R)

is-set-subset-Semiring :
(l : Level) {l1 : Level} (R : Semiring l1) → is-set (subset-Semiring l R)
is-set-subset-Semiring l R =
is-set-function-type is-set-type-Prop

module _
{l1 l2 : Level} (R : Semiring l1) (S : subset-Semiring l2 R)
where

is-in-subset-Semiring : type-Semiring R → UU l2
is-in-subset-Semiring = is-in-subtype S

is-prop-is-in-subset-Semiring :
(x : type-Semiring R) → is-prop (is-in-subset-Semiring x)
is-prop-is-in-subset-Semiring = is-prop-is-in-subtype S

type-subset-Semiring : UU (l1 ⊔ l2)
type-subset-Semiring = type-subtype S

inclusion-subset-Semiring : type-subset-Semiring → type-Semiring R
inclusion-subset-Semiring = inclusion-subtype S

ap-inclusion-subset-Semiring :
(x y : type-subset-Semiring) →
x ＝ y → (inclusion-subset-Semiring x ＝ inclusion-subset-Semiring y)
ap-inclusion-subset-Semiring = ap-inclusion-subtype S

is-in-subset-inclusion-subset-Semiring :
(x : type-subset-Semiring) →
is-in-subset-Semiring (inclusion-subset-Semiring x)
is-in-subset-inclusion-subset-Semiring =
is-in-subtype-inclusion-subtype S

is-closed-under-eq-subset-Semiring :
{x y : type-Semiring R} →
is-in-subset-Semiring x → (x ＝ y) → is-in-subset-Semiring y
is-closed-under-eq-subset-Semiring =
is-closed-under-eq-subtype S

is-closed-under-eq-subset-Semiring' :
{x y : type-Semiring R} →
is-in-subset-Semiring y → (x ＝ y) → is-in-subset-Semiring x
is-closed-under-eq-subset-Semiring' =
is-closed-under-eq-subtype' S


The condition that a subset contains zero

module _
{l1 l2 : Level} (R : Semiring l1) (S : subset-Semiring l2 R)
where

contains-zero-subset-Semiring : UU l2
contains-zero-subset-Semiring = is-in-subtype S (zero-Semiring R)


The condition that a subset contains one

  contains-one-subset-Semiring : UU l2
contains-one-subset-Semiring = is-in-subtype S (one-Semiring R)


The condition that a subset is closed under addition

  is-closed-under-addition-subset-Semiring : UU (l1 ⊔ l2)
(x y : type-Semiring R) →
is-in-subtype S x → is-in-subtype S y →
is-in-subtype S (add-Semiring R x y)


The condition that a subset is closed under multiplication

  is-closed-under-multiplication-subset-Semiring : UU (l1 ⊔ l2)
is-closed-under-multiplication-subset-Semiring =
(x y : type-Semiring R) → is-in-subtype S x → is-in-subtype S y →
is-in-subtype S (mul-Semiring R x y)


The condition that a subset is closed under multiplication from the left by an arbitrary element

  is-closed-under-left-multiplication-subset-Semiring : UU (l1 ⊔ l2)
is-closed-under-left-multiplication-subset-Semiring =
(x y : type-Semiring R) → is-in-subtype S y →
is-in-subtype S (mul-Semiring R x y)


The condition that a subset is closed-under-multiplication from the right by an arbitrary element

  is-closed-under-right-multiplication-subset-Semiring : UU (l1 ⊔ l2)
is-closed-under-right-multiplication-subset-Semiring =
(x y : type-Semiring R) → is-in-subtype S x →
is-in-subtype S (mul-Semiring R x y)