Inverse semigroups

Content created by Fredrik Bakke, Jonathan Prieto-Cubides and Egbert Rijke.

Created on 2022-04-08.
Last modified on 2023-05-28.

module group-theory.inverse-semigroups where
open import foundation.cartesian-product-types
open import foundation.contractible-types
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.sets
open import foundation.universe-levels

open import group-theory.semigroups


An inverse semigroup is an algebraic structure that models partial bijections. In an inverse semigroup, elements have unique inverses in the sense that for each x there is a unique y such that xyx = x and yxy = y.


is-inverse-Semigroup :
  {l : Level} (S : Semigroup l)  UU l
is-inverse-Semigroup S =
  (x : type-Semigroup S) 
    ( Σ ( type-Semigroup S)
        ( λ y 
          Id (mul-Semigroup S (mul-Semigroup S x y) x) x ×
          Id (mul-Semigroup S (mul-Semigroup S y x) y) y))

Inverse-Semigroup : (l : Level)  UU (lsuc l)
Inverse-Semigroup l = Σ (Semigroup l) is-inverse-Semigroup

module _
  {l : Level} (S : Inverse-Semigroup l)

  semigroup-Inverse-Semigroup : Semigroup l
  semigroup-Inverse-Semigroup = pr1 S

  set-Inverse-Semigroup : Set l
  set-Inverse-Semigroup = set-Semigroup semigroup-Inverse-Semigroup

  type-Inverse-Semigroup : UU l
  type-Inverse-Semigroup = type-Semigroup semigroup-Inverse-Semigroup

  is-set-type-Inverse-Semigroup : is-set type-Inverse-Semigroup
  is-set-type-Inverse-Semigroup =
    is-set-type-Semigroup semigroup-Inverse-Semigroup

  mul-Inverse-Semigroup :
    (x y : type-Inverse-Semigroup)  type-Inverse-Semigroup
  mul-Inverse-Semigroup = mul-Semigroup semigroup-Inverse-Semigroup

  mul-Inverse-Semigroup' :
    (x y : type-Inverse-Semigroup)  type-Inverse-Semigroup
  mul-Inverse-Semigroup' = mul-Semigroup' semigroup-Inverse-Semigroup

  associative-mul-Inverse-Semigroup :
    (x y z : type-Inverse-Semigroup) 
      ( mul-Inverse-Semigroup (mul-Inverse-Semigroup x y) z)
      ( mul-Inverse-Semigroup x (mul-Inverse-Semigroup y z))
  associative-mul-Inverse-Semigroup =
    associative-mul-Semigroup semigroup-Inverse-Semigroup

  is-inverse-semigroup-Inverse-Semigroup :
    is-inverse-Semigroup semigroup-Inverse-Semigroup
  is-inverse-semigroup-Inverse-Semigroup = pr2 S

  inv-Inverse-Semigroup : type-Inverse-Semigroup  type-Inverse-Semigroup
  inv-Inverse-Semigroup x =
    pr1 (center (is-inverse-semigroup-Inverse-Semigroup x))

  inner-inverse-law-mul-Inverse-Semigroup :
    (x : type-Inverse-Semigroup) 
      ( mul-Inverse-Semigroup
        ( mul-Inverse-Semigroup x (inv-Inverse-Semigroup x))
        ( x))
      ( x)
  inner-inverse-law-mul-Inverse-Semigroup x =
    pr1 (pr2 (center (is-inverse-semigroup-Inverse-Semigroup x)))

  outer-inverse-law-mul-Inverse-Semigroup :
    (x : type-Inverse-Semigroup) 
      ( mul-Inverse-Semigroup
        ( mul-Inverse-Semigroup (inv-Inverse-Semigroup x) x)
        ( inv-Inverse-Semigroup x))
      ( inv-Inverse-Semigroup x)
  outer-inverse-law-mul-Inverse-Semigroup x =
    pr2 (pr2 (center (is-inverse-semigroup-Inverse-Semigroup x)))

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