# mathlib3documentation

group_theory.submonoid.center

# Centers of monoids #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

## Main definitions #

• submonoid.center: the center of a monoid
• add_submonoid.center: the center of an additive monoid

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

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] :
@[simp]
theorem submonoid.center_to_subsemigroup (M : Type u_1) [monoid M] :
theorem add_submonoid.mem_center_iff {M : Type u_1} [add_monoid M] {z : M} :
(g : M), g + z = z + g
theorem submonoid.mem_center_iff {M : Type u_1} [monoid M] {z : M} :
(g : M), g * z = z * g
@[protected, instance]
def add_submonoid.decidable_mem_center {M : Type u_1} [add_monoid M] (a : M) [decidable ( (b : M), b + a = a + b)] :
Equations
@[protected, instance]
def submonoid.decidable_mem_center {M : Type u_1} [monoid M] (a : M) [decidable ( (b : M), b * a = a * b)] :
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] :