Data.Semiring.Property

Properties of pre-semirings & semirings

neutral_addition_on :: Semigroup r => Rel r -> r -> r -> Bool

\( \forall a \in R: (z + a) \sim a \)

A (pre-)semiring with a right-neutral additive unit must satisfy:

neutral_addition mempty ~~ const True

Or, equivalently:

mempty <> r ~~ r

This is a required property.

neutral_addition_on' :: Monoid r => Rel r -> r -> Bool

neutral_multiplication_on :: Semiring r => Rel r -> r -> r -> Bool

\( \forall a \in R: (o * a) \sim a \)

A (pre-)semiring with a right-neutral multiplicative unit must satisfy:

neutral_multiplication unit ~~ const True

Or, equivalently:

unit >< r ~~ r

This is a required property.

neutral_multiplication_on' :: (Monoid r, Semiring r) => Rel r -> r -> Bool

associative_addition_on :: Semigroup r => Rel r -> r -> r -> r -> Bool

\( \forall a, b, c \in R: (a + b) + c \sim a + (b + c) \)

R must right-associate addition.

This should be verified by the underlying Semigroup instance, but is included here for completeness.

This is a required property.

associative_multiplication_on :: Semiring r => Rel r -> r -> r -> r -> Bool

\( \forall a, b, c \in R: (a * b) * c \sim a * (b * c) \)

R must right-associate multiplication.

This is a required property.

distributive_on :: Semiring r => Rel r -> r -> r -> r -> Bool

\( \forall a, b, c \in R: (a + b) * c \sim (a * c) + (b * c) \)

R must right-distribute multiplication.

When R is a functor and the semiring structure is derived from Alternative, this translates to:

(a <|> b) *> c = (a *> c) <|> (b *> c)

See https://en.wikibooks.org/wiki/Haskell/Alternative_and_MonadPlus.

This is a required property.

Properties of non-unital (near-)semirings

nonunital_on :: (Monoid r, Semiring r) => Rel r -> r -> r -> Bool

\( \forall a, b \in R: a * b \sim a * b + b \)

If R is non-unital (i.e. unit is not distinct from mempty) then it will instead satisfy a right-absorbtion property.

This follows from right-neutrality and right-distributivity.

Compare codistributive and closed_stable.

When R is also left-distributive we get: \( \forall a, b \in R: a * b = a + a * b + b \)

See also Warning and https://blogs.ncl.ac.uk/andreymokhov/united-monoids/#whatif.

Properties of unital semirings

annihilative_multiplication_on :: (Monoid r, Semiring r) => Rel r -> r -> Bool

\( \forall a \in R: (z * a) \sim u \)

A R is unital then its addititive unit must be right-annihilative, i.e.:

mempty >< a ~~ mempty

For Alternative instances this property translates to:

empty *> a ~~ empty

All right semirings must have a right-absorbative addititive unit, however note that depending on the Prd instance this does not preclude IEEE754-mandated behavior such as:

mempty >< NaN ~~ NaN

This is a required property.

homomorphism_boolean :: forall r. (Eq r, Monoid r, Semiring r) => Bool -> Bool -> Bool

fromBoolean must be a semiring homomorphism into R.

This is a required property.

Properties of cancellative semirings

cancellative_addition_on :: Semigroup r => Rel r -> r -> r -> r -> Bool

\( \forall a, b, c \in R: b + a \sim c + a \Rightarrow b = c \)

If R is right-cancellative wrt addition then for all a the section (a <>) is injective.

cancellative_multiplication_on :: Semiring r => Rel r -> r -> r -> r -> Bool

\( \forall a, b, c \in R: b * a \sim c * a \Rightarrow b = c \)

If R is right-cancellative wrt multiplication then for all a the section (a ><) is injective.

Properties of commutative semirings

commutative_addition_on :: Semigroup r => Rel r -> r -> r -> Bool

\( \forall a, b \in R: a + b \sim b + a \)

commutative_multiplication_on :: Semiring r => Rel r -> r -> r -> Bool

\( \forall a, b \in R: a * b \sim b * a \)

Properties of distributive semirings

distributive_finite_on :: (Monoid r, Semiring r) => Rel r -> [r] -> r -> Bool

\( \forall M \geq 0; a_1 \dots a_M, b \in R: (\sum_{i=1}^M a_i) * b \sim \sum_{i=1}^M a_i * b \)

R must right-distribute multiplication between finite sums.

For types with exact arithmetic this follows from distributive & neutral_multiplication.

distributive_finite1_on :: Semiring r => Rel r -> NonEmpty r -> r -> Bool

\( \forall M \geq 1; a_1 \dots a_M, b \in R: (\sum_{i=1}^M a_i) * b \sim \sum_{i=1}^M a_i * b \)

R must right-distribute multiplication over finite (non-empty) sums.

For types with exact arithmetic this follows from distributive and the universality of fold1.

distributive_cross_on :: (Monoid r, Semiring r) => Rel r -> [r] -> [r] -> Bool

\( \forall M,N \geq 0; a_1 \dots a_M, b_1 \dots b_N \in R: (\sum_{i=1}^M a_i) * (\sum_{j=1}^N b_j) \sim \sum_{i=1 j=1}^{i=M j=N} a_i * b_j \)

If R is also left-distributive then it supports cross-multiplication.

distributive_cross1_on :: Semiring r => Rel r -> NonEmpty r -> NonEmpty r -> Bool

\( \forall M,N \geq 1; a_1 \dots a_M, b_1 \dots b_N \in R: (\sum_{i=1}^M a_i) * (\sum_{j=1}^N b_j) = \sum_{i=1 j=1}^{i=M j=N} a_i * b_j \)

If R is also left-distributive then it supports (non-empty) cross-multiplication.