mathlib documentation

algebra.star.basic

Star monoids, rings, and modules #

We introduce the basic algebraic notions of star monoids, star rings, and star modules. A star algebra is simply a star ring that is also a star module.

These are implemented as "mixin" typeclasses, so to summon a star ring (for example) one needs to write (R : Type) [ring R] [star_ring R]. This avoids difficulties with diamond inheritance.

For now we simply do not introduce notations, as different users are expected to feel strongly about the relative merits of r^*, r†, rᘁ, and so on.

Our star rings are actually star semirings, but of course we can prove star_neg : star (-r) = - star r when the underlying semiring is a ring.

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@[class]
structure has_star (R : Type u) :
Type u
  • star : R → R

Notation typeclass (with no default notation!) for an algebraic structure with a star operation.

Instances
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@[class]
structure has_involutive_star (R : Type u) :
Type u

Typeclass for a star operation with is involutive.

Instances
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@[simp]
theorem star_star {R : Type u} [has_involutive_star R] (r : R) :
star (star r) = r
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theorem star_injective {R : Type u} [has_involutive_star R] :
function.injective star
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@[class]
structure star_monoid (R : Type u) [monoid R] :
Type u

A *-monoid is a monoid R with an involutive operations star so star (r * s) = star s * star r.

Instances
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@[simp]
theorem star_mul' {R : Type u} [comm_monoid R] [star_monoid R] (x y : R) :
star (x * y) = (star x) * star y

In a commutative ring, make simp prefer leaving the order unchanged.

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def star_mul_equiv {R : Type u} [monoid R] [star_monoid R] :
R ≃* Rᵐᵒᵖ

star as an mul_equiv from R to Rᵐᵒᵖ

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@[simp]
theorem star_mul_equiv_apply {R : Type u} [monoid R] [star_monoid R] (x : R) :
⇑star_mul_equiv x = mul_opposite.op (star x)
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@[simp]
theorem star_mul_aut_apply {R : Type u} [comm_monoid R] [star_monoid R] (ᾰ : R) :
⇑star_mul_aut ᾰ = star ᾰ
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def star_mul_aut {R : Type u} [comm_monoid R] [star_monoid R] :
mul_aut R

star as a mul_aut for commutative R.

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@[simp]
theorem star_one (R : Type u) [monoid R] [star_monoid R] :
star 1 = 1
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@[simp]
theorem star_pow {R : Type u} [monoid R] [star_monoid R] (x : R) (n : ℕ) :
star (x ^ n) = star x ^ n
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@[simp]
theorem star_inv {R : Type u} [group R] [star_monoid R] (x : R) :
star x⁻¹ = (star x)⁻¹
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@[simp]
theorem star_zpow {R : Type u} [group R] [star_monoid R] (x : R) (z : ℤ) :
star (x ^ z) = star x ^ z
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@[simp]
theorem star_div {R : Type u} [comm_group R] [star_monoid R] (x y : R) :
star (x / y) = star x / star y

When multiplication is commutative, star preserves division.

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@[simp]
theorem star_prod {R : Type u} [comm_monoid R] [star_monoid R] {α : Type u_1} (s : finset α) (f : α → R) :
star (∏ (x : α) in s, f x) = ∏ (x : α) in s, star (f x)
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def star_monoid_of_comm {R : Type u_1} [comm_monoid R] :
star_monoid R

Any commutative monoid admits the trivial *-structure.

See note [reducible non-instances].

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theorem star_id_of_comm {R : Type u_1} [comm_semiring R] {x : R} :
star x = x

Note that since star_monoid_of_comm is reducible, simp can already prove this. -

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@[class]
structure star_add_monoid (R : Type u) [add_monoid R] :
Type u

A *-additive monoid R is an additive monoid with an involutive star operation which preserves addition.

Instances
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@[simp]
theorem star_add_equiv_apply {R : Type u} [add_monoid R] [star_add_monoid R] (ᾰ : R) :
⇑star_add_equiv ᾰ = star ᾰ
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def star_add_equiv {R : Type u} [add_monoid R] [star_add_monoid R] :
R ≃+ R

star as an add_equiv

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@[simp]
theorem star_zero (R : Type u) [add_monoid R] [star_add_monoid R] :
star 0 = 0
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@[simp]
theorem star_neg {R : Type u} [add_group R] [star_add_monoid R] (r : R) :
star (-r) = -star r
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@[simp]
theorem star_sub {R : Type u} [add_group R] [star_add_monoid R] (r s : R) :
star (r - s) = star r - star s
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@[simp]
theorem star_nsmul {R : Type u} [add_comm_monoid R] [star_add_monoid R] (x : R) (n : ℕ) :
star (n • x) = n • star x
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@[simp]
theorem star_zsmul {R : Type u} [add_comm_group R] [star_add_monoid R] (x : R) (n : ℤ) :
star (n • x) = n • star x
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@[simp]
theorem star_sum {R : Type u} [add_comm_monoid R] [star_add_monoid R] {α : Type u_1} (s : finset α) (f : α → R) :
star (∑ (x : α) in s, f x) = ∑ (x : α) in s, star (f x)
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@[class]
structure star_ring (R : Type u) [semiring R] :
Type u

A *-ring R is a (semi)ring with an involutive star operation which is additive which makes R with its multiplicative structure into a *-monoid (i.e. star (r * s) = star s * star r).

Instances
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@[protected, instance]
def star_ring.to_star_add_monoid {R : Type u} [semiring R] [star_ring R] :
star_add_monoid R
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@[simp]
theorem star_ring_equiv_apply {R : Type u} [semiring R] [star_ring R] (x : R) :
⇑star_ring_equiv x = mul_opposite.op (star x)
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def star_ring_equiv {R : Type u} [semiring R] [star_ring R] :
R ≃+* Rᵐᵒᵖ

star as an ring_equiv from R to Rᵐᵒᵖ

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def star_ring_aut {R : Type u} [comm_semiring R] [star_ring R] :
ring_aut R

star as a ring_aut for commutative R. This is used to denote complex conjugation, and is available under the notation conj in the locale complex_conjugate

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theorem star_ring_aut_apply {R : Type u} [comm_semiring R] [star_ring R] {x : R} :
⇑conj x = star x

This is not a simp lemma, since we usually want simp to keep star_ring_aut bundled. For example, for complex conjugation, we don't want simp to turn conj x into the bare function star x automatically since most lemmas are about conj x.

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@[simp]
theorem star_ring_aut_self_apply {R : Type u} [comm_semiring R] [star_ring R] (x : R) :
⇑conj (⇑conj x) = x
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theorem complex.conj_conj {R : Type u} [comm_semiring R] [star_ring R] (x : R) :
⇑conj (⇑conj x) = x

Alias of star_ring_aut_self_apply.

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theorem is_R_or_C.conj_conj {R : Type u} [comm_semiring R] [star_ring R] (x : R) :
⇑conj (⇑conj x) = x

Alias of star_ring_aut_self_apply.

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@[simp]
theorem star_inv' {R : Type u} [division_ring R] [star_ring R] (x : R) :
star x⁻¹ = (star x)⁻¹
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@[simp]
theorem star_zpow₀ {R : Type u} [division_ring R] [star_ring R] (x : R) (z : ℤ) :
star (x ^ z) = star x ^ z
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@[simp]
theorem star_div' {R : Type u} [field R] [star_ring R] (x y : R) :
star (x / y) = star x / star y

When multiplication is commutative, star preserves division.

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@[simp]
theorem star_bit0 {R : Type u} [ring R] [star_ring R] (r : R) :
star (bit0 r) = bit0 (star r)
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@[simp]
theorem star_bit1 {R : Type u} [ring R] [star_ring R] (r : R) :
star (bit1 r) = bit1 (star r)
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def star_ring_of_comm {R : Type u_1} [comm_semiring R] :
star_ring R

Any commutative semiring admits the trivial *-structure.

See note [reducible non-instances].

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@[class]
structure star_ordered_ring (R : Type u) [ordered_semiring R] :
Type u

An ordered *-ring is a ring which is both an ordered ring and a *-ring, and 0 ≤ star r * r for every r.

(In a Banach algebra, the natural ordering is given by the positive cone which is the closure of the sums of elements star r * r. This ordering makes the Banach algebra an ordered *-ring.)

Instances
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theorem star_mul_self_nonneg {R : Type u} [ordered_semiring R] [star_ordered_ring R] {r : R} :
0 ≤ (star r) * r
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@[class]
structure star_module (R : Type u) (A : Type v) [has_star R] [has_star A] [has_scalar R A] :
Type

A star module A over a star ring R is a module which is a star add monoid, and the two star structures are compatible in the sense star (r • a) = star r • star a.

Note that it is up to the user of this typeclass to enforce [semiring R] [star_ring R] [add_comm_monoid A] [star_add_monoid A] [module R A], and that the statement only requires [has_star R] [has_star A] [has_scalar R A].

If used as [comm_ring R] [star_ring R] [semiring A] [star_ring A] [algebra R A], this represents a star algebra.

Instances
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@[protected, instance]
def star_monoid.to_star_module {R : Type u} [comm_monoid R] [star_monoid R] :
star_module R R

A commutative star monoid is a star module over itself via monoid.to_mul_action.

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@[protected, instance]
def ring_hom_inv_pair.star_ring_aut.ring_hom_inv_pair {R : Type u} [comm_semiring R] [star_ring R] :
ring_hom_inv_pair ↑conj ↑conj

Instance needed to define star-linear maps over a commutative star ring (ex: conjugate-linear maps when R = ℂ).

Instances #

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@[protected, instance]
def units.star_monoid {R : Type u} [monoid R] [star_monoid R] :
star_monoid (units R)
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@[simp]
theorem units.coe_star {R : Type u} [monoid R] [star_monoid R] (u : units R) :
↑(star u) = star ↑u
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@[simp]
theorem units.coe_star_inv {R : Type u} [monoid R] [star_monoid R] (u : units R) :
↑(star u)⁻¹ = star ↑u⁻¹
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@[protected, instance]
def units.star_module {R : Type u} [monoid R] [star_monoid R] {A : Type u_1} [has_star A] [has_scalar R A] [star_module R A] :
star_module (units R) A
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theorem is_unit.star {R : Type u} [monoid R] [star_monoid R] {a : R} :
is_unit a → is_unit (star a)
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@[simp]
theorem is_unit_star {R : Type u} [monoid R] [star_monoid R] {a : R} :
is_unit (star a) ↔ is_unit a
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theorem ring.inverse_star {R : Type u} [semiring R] [star_ring R] (a : R) :
ring.inverse (star a) = star (ring.inverse a)
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@[protected, instance]
def mul_opposite.has_star {R : Type u} [has_star R] :
has_star Rᵐᵒᵖ

The opposite type carries the same star operation.

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@[simp]
theorem mul_opposite.unop_star {R : Type u} [has_star R] (r : Rᵐᵒᵖ) :
mul_opposite.unop (star r) = star (mul_opposite.unop r)
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@[simp]
theorem mul_opposite.op_star {R : Type u} [has_star R] (r : R) :
mul_opposite.op (star r) = star (mul_opposite.op r)
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@[protected, instance]
def mul_opposite.has_involutive_star {R : Type u} [has_involutive_star R] :
has_involutive_star Rᵐᵒᵖ
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@[protected, instance]
def mul_opposite.star_monoid {R : Type u} [monoid R] [star_monoid R] :
star_monoid Rᵐᵒᵖ
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@[protected, instance]
def mul_opposite.star_add_monoid {R : Type u} [add_monoid R] [star_add_monoid R] :
star_add_monoid Rᵐᵒᵖ
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@[protected, instance]
def mul_opposite.star_ring {R : Type u} [semiring R] [star_ring R] :
star_ring Rᵐᵒᵖ
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@[protected, instance]
def star_monoid.to_opposite_star_module {R : Type u} [comm_monoid R] [star_monoid R] :
star_module Rᵐᵒᵖ R

A commutative star monoid is a star module over its opposite via monoid.to_opposite_mul_action.

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