The Fortress Language Specification - CiteSeerX

extends { UnaryOperatorT, ⊙, BinaryPredicateT, ≃ }
property ∀(a:T, a ′ : T) (a ≃ a ′ ) →:(⊙a) ≃ (⊙a ′ )
end
This peculiarly spiffy trait states that the unary operator ⊙ is consistent under substitutions described by the relation
≃ (which is typically, but not always, an equivalence relation); that is, the result produced by ⊙ is unchanged if its
argument is replaced by some other value that is equivalent.
trait BinaryOperatorSubstitutionLawsT extends BinaryOperatorSubstitutionLawsT, ⊙, =,
opr ⊙,opr ≃
extends { BinaryOperatorT, ⊙, BinaryPredicateT, ≃ }
property ∀(a:T, a ′ : T) (a ≃ a ′ ) →: ∀(b: T) (a ⊙ b) ≃ (a ′ ⊙ b)
property ∀(b: T, b ′ : T) (b ≃ b ′ ) →: ∀(a: T) (a ⊙ b) ≃ (a ⊙ b ′ )
end
This equally spiffy trait states that the binary operator ⊙ is consistent under substitutions described by the relation ≃
(which is typically, but not always, an equivalence relation); that is, the result produced by ⊙ is unchanged if either
argument is replaced by some other value that is equivalent. (It is then easy to prove that the result is unchanged even
when both arguments are replaced by equivalent values.)
37.4 Monoids, Groups, Rings, and Fields
trait ApproximateMonoidT extends ApproximateMonoidT, ⊙, ≈,opr ⊙,opr ≈
extends { ApproximatelyAssociativeT, ⊙, ≈, HasIdentityT, ⊙ }
end
An approximate monoid is a set of values with an approximately associative binary operator ⊙ that has an identity.
For example, floating-point multiplication has identity 1 and is approximately associative.
trait MonoidT extends MonoidT, ⊙,opr ⊙
extends { ApproximateMonoidT, ⊙, =, AssociativeT, ⊙ }
end
A monoid is a set of values with an associative binary operator ⊙ that has an identity. For example, trait String
extends MonoidString, ‖ where ‖ is the string concatenation operator. Note that string concatenation is associative
but not commutative and that the empty string is the identity for string concatenation, so coercing Identity ‖, to
type String produces the empty string.
trait ApproximateCommutativeMonoid T extends ApproximateCommutativeMonoidT, ⊕, ≈,
opr ⊕,opr ≈
extends { ApproximateMonoidT, ⊕, ≈, ApproximatelyCommutativeT, ⊕, ≈ }
end
An approximate commutative monoid is an approximate monoid whose binary operator is also approximately commutative.
For example, floating-point multiplication has identity 1 and is approximately associative and also approximately
(indeed, exactly) commutative.
trait CommutativeMonoidT extends CommutativeMonoidT, ⊕,opr ⊕
extends { ApproximateCommutativeMonoidT, ⊕, =, MonoidT, ⊕, CommutativeT, ⊕ }
end
A commutative monoid is a monoid whose binary operator is also commutative. For example, the operator ∧ on
boolean values is associative and commutative and has identity true ; likewise, the operator ∨ on boolean values is
associative and commutative and has identity false .
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