nutcracker.util.algebraic

Monoid with a property that the result of the monoid operation is greater than or equal to any of the operands:

∀ a,b: a ≤ a⊕b b ≤ a⊕b

This property implies that zero is the minimum element:

∀a: 0 ≤ a

Semigroup with a property that the result of the semigroup operation is greater than or equal to any of the operands:

∀ a,b: a ≤ a⊕b b ≤ a⊕b

Monoid with a property that the result of the monoid operation is less than or equal to any of the operands:

∀ a,b: a⊕b ≤ a a⊕b ≤ b

This property implies that zero is the maximum element:

∀a: a ≤ 0

Semigroup with a property that the result of the semigroup operation is less than or equal to any of the operands:

∀ a,b: a⊕b ≤ a a⊕b ≤ b

Monoid whose operation preserves the Order:

∀ a,b,c: (a ≤ b) → (a⊕c ≤ b⊕c) (a ≤ b) → (c⊕a ≤ c⊕b)

Semigroup whose operation preserves the Order:

∀ a,b,c: (a ≤ b) → (a⊕c ≤ b⊕c) (a ≤ b) → (c⊕a ≤ c⊕b)

An auxiliary trait that extends both Semigroup and Order, but does not add any new laws.