Introduce SMulWithZero

In analogy with the usual monoid action on a Type M, we introduce an action of a MonoidWithZero on a Type with 0.

In particular, for Types R and M, both containing 0, we define SMulWithZero R M to be the typeclass where the products r • 0 and 0 • m vanish for all r : R and all m : M.

Moreover, in the case in which R is a MonoidWithZero, we introduce the typeclass MulActionWithZero R M, mimicking group actions and having an absorbing 0 in R. Thus, the action is required to be compatible with

• the unit of the monoid, acting as the identity;
• the zero of the MonoidWithZero, acting as zero;
• associativity of the monoid.

We also add an instance:

• any MonoidWithZero has a MulActionWithZero R R acting on itself.

Main declarations

• smulMonoidWithZeroHom: Scalar multiplication bundled as a morphism of monoids with zero.

class SMulWithZero (R : Type u_1) (M : Type u_3) [Zero R] [Zero M] extends SMulZeroClass R M : Type (max u_1 u_3)

SMulWithZero is a class consisting of a Type R with 0 ∈ R and a scalar multiplication of R on a Type M with 0, such that the equality r • m = 0 holds if at least one among r or m equals 0.

• smul : R → M → M
• smul_zero : ∀ (a : R), a • 0 = 0
• zero_smul : ∀ (m : M), 0 • m = 0

  Scalar multiplication by the scalar 0 is 0.

instance MulZeroClass.toSMulWithZero (R : Type u_1) [MulZeroClass R] : SMulWithZero R R

Equations

• MulZeroClass.toSMulWithZero R = SMulWithZero.mk ⋯

instance MulZeroClass.toOppositeSMulWithZero (R : Type u_1) [MulZeroClass R] : SMulWithZero Rᵐᵒᵖ R

Like MulZeroClass.toSMulWithZero, but multiplies on the right.

Equations

• MulZeroClass.toOppositeSMulWithZero R = SMulWithZero.mk ⋯

@[simp]

theorem zero_smul (R : Type u_1) {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (m : M) : 0 • m = 0

theorem smul_eq_zero_of_left {R : Type u_1} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] {a : R} (h : a = 0) (b : M) : a • b = 0

theorem left_ne_zero_of_smul {R : Type u_1} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] {a : R} {b : M} : a • b ≠ 0 → a ≠ 0

@[reducible, inline]

abbrev Function.Injective.smulWithZero {R : Type u_1} {M : Type u_3} {M' : Type u_4} [Zero R] [Zero M] [SMulWithZero R M] [Zero M'] [SMul R M'] (f : ZeroHom M' M) (hf : Function.Injective ⇑f) (smul : ∀ (a : R) (b : M'), f (a • b) = a • f b) : SMulWithZero R M'

Pullback a SMulWithZero structure along an injective zero-preserving homomorphism. See note [reducible non-instances].

Equations

• Function.Injective.smulWithZero f hf smul = SMulWithZero.mk ⋯

@[reducible, inline]

abbrev Function.Surjective.smulWithZero {R : Type u_1} {M : Type u_3} {M' : Type u_4} [Zero R] [Zero M] [SMulWithZero R M] [Zero M'] [SMul R M'] (f : ZeroHom M M') (hf : Function.Surjective ⇑f) (smul : ∀ (a : R) (b : M), f (a • b) = a • f b) : SMulWithZero R M'

Pushforward a SMulWithZero structure along a surjective zero-preserving homomorphism. See note [reducible non-instances].

Equations

• Function.Surjective.smulWithZero f hf smul = SMulWithZero.mk ⋯

def SMulWithZero.compHom {R : Type u_1} {R' : Type u_2} (M : Type u_3) [Zero R] [Zero M] [SMulWithZero R M] [Zero R'] (f : ZeroHom R' R) : SMulWithZero R' M

Compose a SMulWithZero with a ZeroHom, with action f r' • m

Equations

• SMulWithZero.compHom M f = SMulWithZero.mk ⋯

instance AddMonoid.natSMulWithZero {M : Type u_3} [AddMonoid M] : SMulWithZero ℕ M

Equations

• AddMonoid.natSMulWithZero = SMulWithZero.mk ⋯

instance AddGroup.intSMulWithZero {M : Type u_3} [AddGroup M] : SMulWithZero ℤ M

Equations

• AddGroup.intSMulWithZero = SMulWithZero.mk ⋯

class MulActionWithZero (R : Type u_1) (M : Type u_3) [MonoidWithZero R] [Zero M] extends MulAction R M : Type (max u_1 u_3)

An action of a monoid with zero R on a Type M, also with 0, extends MulAction and is compatible with 0 (both in R and in M), with 1 ∈ R, and with associativity of multiplication on the monoid M.

• smul : R → M → M
• one_smul : ∀ (b : M), 1 • b = b
• mul_smul : ∀ (x y : R) (b : M), (x * y) • b = x • y • b
• smul_zero : ∀ (r : R), r • 0 = 0

  Scalar multiplication by any element send 0 to 0.

• zero_smul : ∀ (m : M), 0 • m = 0

  Scalar multiplication by the scalar 0 is 0.

@[instance 100]

instance MulActionWithZero.toSMulWithZero (R : Type u_5) (M : Type u_6) {x✝ : MonoidWithZero R} {x✝¹ : Zero M} [m : MulActionWithZero R M] : SMulWithZero R M

Equations

• MulActionWithZero.toSMulWithZero R M = SMulWithZero.mk ⋯

instance MonoidWithZero.toMulActionWithZero (R : Type u_1) [MonoidWithZero R] : MulActionWithZero R R

See also Semiring.toModule

Equations

• MonoidWithZero.toMulActionWithZero R = MulActionWithZero.mk ⋯ ⋯

instance MonoidWithZero.toOppositeMulActionWithZero (R : Type u_1) [MonoidWithZero R] : MulActionWithZero Rᵐᵒᵖ R

Like MonoidWithZero.toMulActionWithZero, but multiplies on the right. See also Semiring.toOppositeModule

Equations

• MonoidWithZero.toOppositeMulActionWithZero R = MulActionWithZero.mk ⋯ ⋯

theorem MulActionWithZero.subsingleton (R : Type u_1) (M : Type u_3) [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Subsingleton R] : Subsingleton M

theorem MulActionWithZero.nontrivial (R : Type u_1) (M : Type u_3) [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Nontrivial M] : Nontrivial R

theorem ite_zero_smul {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] (p : Prop) [Decidable p] (a : R) (b : M) : (if p then a else 0) • b = if p then a • b else 0

theorem boole_smul {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] (p : Prop) [Decidable p] (a : M) : (if p then 1 else 0) • a = if p then a else 0

theorem Pi.single_apply_smul {R : Type u_1} {M : Type u_3} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] {ι : Type u_5} [DecidableEq ι] (x : M) (i j : ι) : Pi.single i 1 j • x = Pi.single i x j

@[reducible, inline]

abbrev Function.Injective.mulActionWithZero {R : Type u_1} {M : Type u_3} {M' : Type u_4} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Zero M'] [SMul R M'] (f : ZeroHom M' M) (hf : Function.Injective ⇑f) (smul : ∀ (a : R) (b : M'), f (a • b) = a • f b) : MulActionWithZero R M'

Pullback a MulActionWithZero structure along an injective zero-preserving homomorphism. See note [reducible non-instances].

Equations

• Function.Injective.mulActionWithZero f hf smul = MulActionWithZero.mk ⋯ ⋯

@[reducible, inline]

abbrev Function.Surjective.mulActionWithZero {R : Type u_1} {M : Type u_3} {M' : Type u_4} [MonoidWithZero R] [Zero M] [MulActionWithZero R M] [Zero M'] [SMul R M'] (f : ZeroHom M M') (hf : Function.Surjective ⇑f) (smul : ∀ (a : R) (b : M), f (a • b) = a • f b) : MulActionWithZero R M'

Pushforward a MulActionWithZero structure along a surjective zero-preserving homomorphism. See note [reducible non-instances].

Equations

• Function.Surjective.mulActionWithZero f hf smul = MulActionWithZero.mk ⋯ ⋯

def MulActionWithZero.compHom {R : Type u_1} {R' : Type u_2} (M : Type u_3) [MonoidWithZero R] [MonoidWithZero R'] [Zero M] [MulActionWithZero R M] (f : R' →*₀ R) : MulActionWithZero R' M

Compose a MulActionWithZero with a MonoidWithZeroHom, with action f r' • m

Equations

• MulActionWithZero.compHom M f = MulActionWithZero.mk ⋯ ⋯

theorem smul_inv₀ {α : Type u_5} {β : Type u_6} [GroupWithZero α] [GroupWithZero β] [MulActionWithZero α β] [SMulCommClass α β β] [IsScalarTower α β β] (c : α) (x : β) : (c • x)⁻¹ = c⁻¹ • x⁻¹