Type Members

1.  trait NonDecreasingMonoid[A] extends NonDecreasingSemigroup[A] with Monoid[A]

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

   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

2.  trait NonDecreasingSemigroup[A] extends SemigroupWithOrder[A]

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

   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

3.  trait NonIncreasingMonoid[A] extends NonIncreasingSemigroup[A] with Monoid[A]

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

   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

4.  trait NonIncreasingSemigroup[A] extends SemigroupWithOrder[A]

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

   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

5.  trait OrderPreservingMonoid[A] extends OrderPreservingSemigroup[A] with Monoid[A]

   Monoid whose operation preserves the Order:

   Monoid whose operation preserves the Order:

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

   See also

   https://en.wikipedia.org/wiki/Ordered_semigroup

6.  trait OrderPreservingSemigroup[A] extends SemigroupWithOrder[A]

   Semigroup whose operation preserves the Order:

   Semigroup whose operation preserves the Order:

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

   See also

   https://en.wikipedia.org/wiki/Ordered_semigroup

7.  trait SemigroupWithOrder[A] extends Semigroup[A] with Order[A]

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