mathlib documentation

group_theory.submonoid.center

Centers of monoids #

Main definitions #

We provide subgroup.center, add_subgroup.center, subsemiring.center, and subring.center in other files.

def submonoid.center (M : Type u_1) [monoid M] :

The center of a monoid M is the set of elements that commute with everything in M

Equations
Instances for submonoid.center
def add_submonoid.center (M : Type u_1) [add_monoid M] :

The center of a monoid M is the set of elements that commute with everything in M

Equations
theorem submonoid.coe_center (M : Type u_1) [monoid M] :
theorem add_submonoid.mem_center_iff {M : Type u_1} [add_monoid M] {z : M} :
z add_submonoid.center M ∀ (g : M), g + z = z + g
theorem submonoid.mem_center_iff {M : Type u_1} [monoid M] {z : M} :
z submonoid.center M ∀ (g : M), g * z = z * g
@[protected, instance]
def submonoid.decidable_mem_center {M : Type u_1} [monoid M] [decidable_eq M] [fintype M] :
decidable_pred (λ (_x : M), _x submonoid.center M)
Equations
@[protected, instance]

The center of a monoid is commutative.

Equations
@[protected, instance]

The center of a monoid acts commutatively on that monoid.

@[protected, instance]

The center of a monoid acts commutatively on that monoid.

Note that smul_comm_class (center M) (center M) M is already implied by submonoid.smul_comm_class_right

@[simp]
theorem submonoid.center_eq_top (M : Type u_1) [comm_monoid M] :