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:

• quotient group: src/group_theory/quotient_group.lean
• quotient module: src/linear_algebra/quotient.lean
• quotient ring: src/ring_theory/ideal/quotient.lean

## Notations #

The following notation is introduced:

• G ⧸ H stands for the quotient of the type G by some term H (for example, H can be a normal subgroup of G). To implement this notation for other quotients, you should provide a has_quotient instance. Note that since G can usually be inferred from H, _ ⧸ H can also be used, but this is less readable.

## Tags #

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

@[class]
structure has_quotient (A : out_param (Type u)) (B : Type v) :
Type (max (u+1) (v+1))
• quotient' : B → Type (max ? ?)

has_quotient 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 of this typeclass
Instances of other typeclasses for has_quotient
• has_quotient.has_sizeof_inst
@[nolint]
def has_quotient.quotient (A : out_param (Type u)) {B : Type v} [ B] (b : B) :
Type (max u v)

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

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

Equations