Documentation

Mathlib.Algebra.Quotient

Algebraic quotients #

This file defines notation for algebraic quotients, e.g. quotient groups G ⧸ H, quotient modules M ⧸ N and ideal quotients R ⧸ I.

The actual quotient structures are defined in the following files:

Notations #

The following notation is introduced:

Tags #

quotient, group quotient, quotient group, module quotient, quotient module, ring quotient, ideal quotient, quotient ring

class HasQuotient (A : outParam (Type u)) (B : Type v) :
Type (max(u+1)(v+1))
  • auxiliary quotient function, the one used will have A explicit

    quotient' : BType (maxuv)

HasQuotient A B is a notation typeclass that allows us to write A ⧸ b for b : B. This allows the usual notation for quotients of algebraic structures, such as groups, modules and rings.

A is a parameter, despite being unused in the definition below, so it appears in the notation.

Instances
    def HasQuotient.Quotient (A : outParam (Type u)) {B : Type v} [inst : HasQuotient A B] (b : B) :
    Type (maxuv)

    HasQuotient.Quotient A b (with notation A ⧸ b) is the quotient of the type A by b.

    This differs from HasQuotient.quotient' in that the A argument is explicit, which is necessary to make Lean show the notation in the goal state.

    Equations