mathlib documentation

group_theory.submonoid.centralizer

Centralizers of magmas and monoids #

Main definitions #

We provide subgroup.centralizer, add_subgroup.centralizer in other files.

def submonoid.centralizer {M : Type u_1} (S : set M) [monoid M] :

The centralizer of a subset of a monoid M.

Equations
def add_submonoid.centralizer {M : Type u_1} (S : set M) [add_monoid M] :

The centralizer of a subset of an additive monoid.

Equations
@[simp, norm_cast]
@[simp, norm_cast]
theorem submonoid.coe_centralizer {M : Type u_1} (S : set M) [monoid M] :
theorem submonoid.mem_centralizer_iff {M : Type u_1} {S : set M} [monoid M] {z : M} :
z submonoid.centralizer S ∀ (g : M), g Sg * z = z * g
theorem add_submonoid.mem_centralizer_iff {M : Type u_1} {S : set M} [add_monoid M] {z : M} :
z add_submonoid.centralizer S ∀ (g : M), g Sg + z = z + g
@[protected, instance]
def submonoid.decidable_mem_centralizer {M : Type u_1} {S : set M} [monoid M] [decidable_eq M] [fintype M] [decidable_pred (λ (_x : M), _x S)] :
Equations
@[protected, instance]
def add_submonoid.decidable_mem_centralizer {M : Type u_1} {S : set M} [add_monoid M] [decidable_eq M] [fintype M] [decidable_pred (λ (_x : M), _x S)] :
Equations
theorem submonoid.centralizer_le {M : Type u_1} {S T : set M} [monoid M] (h : S T) :