Documentation

Mathlib.Algebra.SMulWithZero

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

We also add an instance:

Main declarations #

class SMulWithZero (R : Type u_1) (M : Type u_3) [Zero R] [Zero M] extends SMulZeroClass :
Type (max u_1 u_3)
  • smul : RMM
  • smul_zero : ∀ (a : R), a 0 = 0
  • zero_smul : ∀ (m : M), 0 m = 0

    Scalar multiplication by the scalar 0 is 0.

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.

Instances
    Equations

    Like MulZeroClass.toSMulWithZero, but multiplies on the right.

    Equations
    @[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 smul_eq_zero_of_right {R : Type u_1} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] {b : M} (a : R) (h : b = 0) :
    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 0a 0
    theorem right_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 0b 0
    @[reducible]
    def 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) :

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

    Equations
    Instances For
      @[reducible]
      def 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) :

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

      Equations
      Instances For
        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) :

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

        Equations
        Instances For
          Equations
          Equations
          class MulActionWithZero (R : Type u_1) (M : Type u_3) [MonoidWithZero R] [Zero M] extends MulAction :
          Type (max u_1 u_3)
          • smul : RMM
          • 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.

          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.

          Instances
            instance MulActionWithZero.toSMulWithZero (R : Type u_1) (M : Type u_3) [MonoidWithZero R] [Zero M] [m : MulActionWithZero R M] :
            Equations

            See also Semiring.toModule

            Equations

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

            Equations
            • One or more equations did not get rendered due to their size.
            @[reducible]
            def 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) :

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

            Equations
            • One or more equations did not get rendered due to their size.
            Instances For
              @[reducible]
              def 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) :

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

              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
                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) :

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

                Equations
                Instances For
                  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) :
                  @[simp]
                  theorem smulMonoidWithZeroHom_apply {α : Type u_5} {β : Type u_6} [MonoidWithZero α] [MulZeroOneClass β] [MulActionWithZero α β] [IsScalarTower α β β] [SMulCommClass α β β] :
                  ∀ (a : α × β), smulMonoidWithZeroHom a = OneHom.toFun (smulMonoidHom) a
                  def smulMonoidWithZeroHom {α : Type u_5} {β : Type u_6} [MonoidWithZero α] [MulZeroOneClass β] [MulActionWithZero α β] [IsScalarTower α β β] [SMulCommClass α β β] :
                  α × β →*₀ β

                  Scalar multiplication as a monoid homomorphism with zero.

                  Equations
                  • One or more equations did not get rendered due to their size.
                  Instances For