Structures · Algebra
CommGroupWithZero
A type G₀ is a commutative “group with zero”
if it is a commutative monoid with zero element (distinct from 1)
such that every nonzero element is invertible.
The type is required to come with an “inverse” function, and the inverse of 0 must be 0.
- Defined in
- Mathlib.Algebra.GroupWithZero.Defs
- Shape
- One type argument · adds div_eq_mul_inv, zpow_zero', zpow_succ', zpow_neg', inv_zero, mul_inv_cancel
Extends2
Extended by2
Forgetful instances
Every CommGroupWithZero is also a
Concrete types that are instances11
- Rat
- DirectLimit
- OreLocalization
- SignType
- ValuationRing.ValueGroup
- MonoidWithZeroHom.ValueGroup₀
- Subtype
- OrderDual
- ULift
- Lex
- WithZero
How is a type an instance?
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Assumed by138
- mul_div_cancel₀
- le_div_iff₀'
- div_le_iff₀'
- mul_div_mul_left
- div_eq_div_iff
- div_lt_iff₀'
- lt_div_iff₀'
- div_le_div_iff₀
- Set.preimage_const_mul_Ioi₀
- RatFunc.liftMonoidWithZeroHom
- div_div_cancel₀
- div_lt_div_iff₀
- RatFunc.liftMonoidWithZeroHom_apply_div
- RatFunc.liftMonoidWithZeroHom_apply_ofFractionRing_mk
- MulChar.inv_apply'
- CommGroupWithZero.toInv
- div_mul_cancel_left₀
- div_mul_div_cancel₀'
- CommGroupWithZero.coe_normUnit
- Set.preimage_const_mul_Iio₀
- inv_mul_lt_iff₀'
- CommGroupWithZero.toZPow
- MonoidWithZeroHom.ValueGroup₀.zero_or_exists_mk
- HasProd.inv₀
- MonoidWithZeroHom.mem_valueGroup_iff_of_comm
- div_mul_eq_mul_div₀
- HasProd.congr_cofinite₀
- Set.preimage_const_mul_Iic₀
- Units.mk0_prod
- RatFunc.liftMonoidWithZeroHom_injective
- RatFunc.liftMonoidWithZeroHom_apply_div'
- mul_inv_lt_iff₀'
- star_div₀
- Set.preimage_const_mul_Ici₀
- lt_inv_mul_iff₀'
- HasProd.div₀
- Equiv.divLeft₀
- mul_inv_le_iff₀'
- RatFunc.liftMonoidWithZeroHom_apply
- CommGroupWithZero.toDiv
- MulChar.inv_apply_eq_inv'
- divMonoidWithZeroHom
- MonoidWithZeroHom.valueGroup.mk.congr_simp
- Set.preimage_const_mul_Ioo₀
- Multipliable.congr_cofinite₀
- div_eq_div_iff_div_eq_div'
- MonoidWithZeroHom.mem_valueGroup_iff_of_comm'
- le_inv_mul_iff₀'
- div_div_div_cancel_left'
- MonoidWithZeroHom.mker_inverse
Ancestors38
- CommMagma
- CommMonoid
- CommMonoidWithZero
- CommSemigroup
- Div
- DivInvMonoid
- DivInvOneMonoid
- DivisionCommMonoid
- DivisionMonoid
- Dvd
- GroupWithZero
- HDiv
- HMul
- HSMul
- Inv
- InvOneClass
- InvolutiveInv
- Monoid
- MonoidWithZero
- Mul
- MulAction
- MulOne
- MulOneClass
- MulZeroClass
- MulZeroOneClass
- NPow
- NSMul
- Nonempty
- Nontrivial
- OfNat
- One
- SMul
- Semigroup
- SemigroupAction
- SemigroupWithZero
- ZPow
- ZSMul
- Zero