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.

We also define the class star_ordered_ring R, which says that the order on R respects the star operation, i.e. an element r is nonnegative iff there exists an s such that r = star s * s.

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.

TODO #

@[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 of this typeclass
Instances of other typeclasses for has_star
  • has_star.has_sizeof_inst
@[class]
structure has_involutive_star (R : Type u) :
Type u

Typeclass for a star operation with is involutive.

Instances of this typeclass
Instances of other typeclasses for has_involutive_star
  • has_involutive_star.has_sizeof_inst
@[simp]
theorem star_star {R : Type u} [has_involutive_star R] (r : R) :
@[protected]
def equiv.star {R : Type u} [has_involutive_star R] :

star as an equivalence when it is involutive.

Equations
theorem eq_star_of_eq_star {R : Type u} [has_involutive_star R] {r s : R} (h : r = has_star.star s) :
theorem eq_star_iff_eq_star {R : Type u} [has_involutive_star R] {r s : R} :
theorem star_eq_iff_star_eq {R : Type u} [has_involutive_star R] {r s : R} :
@[class]
structure has_trivial_star (R : Type u) [has_star R] :
Prop

Typeclass for a trivial star operation. This is mostly meant for .

Instances of this typeclass
@[class]
structure star_semigroup (R : Type u) [semigroup R] :
Type u

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

Instances of this typeclass
Instances of other typeclasses for star_semigroup
  • star_semigroup.has_sizeof_inst
@[simp]
theorem star_mul' {R : Type u} [comm_semigroup R] [star_semigroup R] (x y : R) :

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

def star_mul_equiv {R : Type u} [semigroup R] [star_semigroup R] :

star as an mul_equiv from R to Rᵐᵒᵖ

Equations
@[simp]
theorem star_mul_aut_apply {R : Type u} [comm_semigroup R] [star_semigroup R] (ᾰ : R) :
def star_mul_aut {R : Type u} [comm_semigroup R] [star_semigroup R] :

star as a mul_aut for commutative R.

Equations
@[simp]
theorem star_one (R : Type u) [monoid R] [star_semigroup R] :
@[simp]
theorem star_pow {R : Type u} [monoid R] [star_semigroup R] (x : R) (n : ) :
@[simp]
theorem star_inv {R : Type u} [group R] [star_semigroup R] (x : R) :
@[simp]
theorem star_zpow {R : Type u} [group R] [star_semigroup R] (x : R) (z : ) :
@[simp]
theorem star_div {R : Type u} [comm_group R] [star_semigroup R] (x y : R) :

When multiplication is commutative, star preserves division.

@[simp]
theorem star_prod {R : Type u} [comm_monoid R] [star_semigroup R] {α : Type u_1} (s : finset α) (f : α → R) :
has_star.star (s.prod (λ (x : α), f x)) = s.prod (λ (x : α), has_star.star (f x))
def star_semigroup_of_comm {R : Type u_1} [comm_monoid R] :

Any commutative monoid admits the trivial *-structure.

See note [reducible non-instances].

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

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

@[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 of this typeclass
Instances of other typeclasses for star_add_monoid
  • star_add_monoid.has_sizeof_inst
@[simp]
theorem star_add_equiv_apply {R : Type u} [add_monoid R] [star_add_monoid R] (ᾰ : R) :
@[simp]
theorem star_zero (R : Type u) [add_monoid R] [star_add_monoid R] :
@[simp]
theorem star_eq_zero {R : Type u} [add_monoid R] [star_add_monoid R] {x : R} :
theorem star_ne_zero {R : Type u} [add_monoid R] [star_add_monoid R] {x : R} :
@[simp]
theorem star_neg {R : Type u} [add_group R] [star_add_monoid R] (r : R) :
@[simp]
theorem star_sub {R : Type u} [add_group R] [star_add_monoid R] (r s : R) :
@[simp]
theorem star_nsmul {R : Type u} [add_monoid R] [star_add_monoid R] (x : R) (n : ) :
@[simp]
theorem star_zsmul {R : Type u} [add_group R] [star_add_monoid R] (x : R) (n : ) :
@[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) :
has_star.star (s.sum (λ (x : α), f x)) = s.sum (λ (x : α), has_star.star (f x))
@[class]
structure star_ring (R : Type u) [non_unital_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 *-semigroup (i.e. star (r * s) = star s * star r).

Instances of this typeclass
Instances of other typeclasses for star_ring
  • star_ring.has_sizeof_inst

star as an ring_equiv from R to Rᵐᵒᵖ

Equations
@[simp, norm_cast]
theorem star_nat_cast {R : Type u} [semiring R] [star_ring R] (n : ) :
@[simp, norm_cast]
theorem star_int_cast {R : Type u} [ring R] [star_ring R] (z : ) :
@[simp, norm_cast]
theorem star_rat_cast {R : Type u} [division_ring R] [char_zero R] [star_ring R] (r : ) :
@[simp]
theorem star_ring_aut_apply {R : Type u} [comm_semiring R] [star_ring R] (ᾰ : R) :
def star_ring_aut {R : Type u} [comm_semiring R] [star_ring R] :

star as a ring automorphism, for commutative R.

Equations
def star_ring_end (R : Type u) [comm_semiring R] [star_ring R] :
R →+* R

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

Note that this is the preferred form (over star_ring_aut, available under the same hypotheses) because the notation E →ₗ⋆[R] F for an R-conjugate-linear map (short for E →ₛₗ[star_ring_end R] F) does not pretty-print if there is a coercion involved, as would be the case for (↑star_ring_aut : R →* R).

Equations
Instances for star_ring_end
theorem star_ring_end_apply {R : Type u} [comm_semiring R] [star_ring R] {x : R} :

This is not a simp lemma, since we usually want simp to keep star_ring_end 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.

@[simp]
theorem star_ring_end_self_apply {R : Type u} [comm_semiring R] [star_ring R] (x : R) :
theorem complex.conj_conj {R : Type u} [comm_semiring R] [star_ring R] (x : R) :

Alias of star_ring_end_self_apply`.

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

Alias of star_ring_end_self_apply`.

@[simp]
theorem star_inv' {R : Type u} [division_ring R] [star_ring R] (x : R) :
@[simp]
theorem star_zpow₀ {R : Type u} [division_ring R] [star_ring R] (x : R) (z : ) :
@[simp]
theorem star_div' {R : Type u} [field R] [star_ring R] (x y : R) :

When multiplication is commutative, star preserves division.

@[simp]
theorem star_bit0 {R : Type u} [add_monoid R] [star_add_monoid R] (r : R) :
@[simp]
theorem star_bit1 {R : Type u} [semiring R] [star_ring R] (r : R) :
def star_ring_of_comm {R : Type u_1} [comm_semiring R] :

Any commutative semiring admits the trivial *-structure.

See note [reducible non-instances].

Equations
@[class]
structure star_ordered_ring (R : Type u) [non_unital_semiring R] [partial_order R] :
Type u

An ordered *-ring is a ring which is both an ordered_add_comm_group and a *-ring, and 0 ≤ r ↔ ∃ s, r = star s * s.

Instances of this typeclass
Instances of other typeclasses for star_ordered_ring
  • star_ordered_ring.has_sizeof_inst
@[protected, instance]
Equations
@[class]
structure star_module (R : Type u) (A : Type v) [has_star R] [has_star A] [has_scalar R A] :
Prop

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 of this typeclass
@[protected, instance]

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

@[protected, instance]

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

Instances #

@[protected, instance]
Equations
@[simp]
theorem units.coe_star {R : Type u} [monoid R] [star_semigroup R] (u : Rˣ) :
@[simp]
@[protected, instance]
def units.star_module {R : Type u} [monoid R] [star_semigroup R] {A : Type u_1} [has_star A] [has_scalar R A] [star_module R A] :
theorem is_unit.star {R : Type u} [monoid R] [star_semigroup R] {a : R} :
@[simp]
theorem is_unit_star {R : Type u} [monoid R] [star_semigroup R] {a : R} :
@[protected, instance]
def invertible.star {R : Type u_1} [monoid R] [star_semigroup R] (r : R) [invertible r] :
Equations
theorem star_inv_of {R : Type u_1} [monoid R] [star_semigroup R] (r : R) [invertible r] [invertible (has_star.star r)] :
@[protected, instance]

The opposite type carries the same star operation.

Equations
@[simp]
@[protected, instance]

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