The Fortress Language Specification - CiteSeerX

trait ApproximateSemiRing T extends ApproximateSemiRingT, ⊕, ⊗, ≈,
opr ⊕,opr ⊗,opr ≈
extends { ApproximateCommutativeMonoidT, ⊕, ≈,
ApproximateMonoidT, ⊗, ≈,
ApproximatelyDistributiveT, ⊗, ⊕, ≈,
ApproximateZeroAnnihilationT, ⊗, ≈ }
property castT(Zero⊗) ≈ castT(Identity⊕)
end
An approximate semiring is a set of values that has two binary operators ⊕ and ⊗ , such that (a) the values form an
approximate commutative monoid with ⊕ ; (b) the values form an approximate monoid with ⊗ ; (c) ⊗ is approximately
distributive over ⊕ ; and (d) ⊗ obeys approximate zero annihilation, where the zero for ⊗ is the same as the
identity for ⊕ .
trait SemiRingT extends SemiRingT, ⊕, ⊗,opr ⊕,opr ⊗
extends { ApproximateSemiRingT, ⊕, ⊗, =,
CommutativeMonoidT, ⊕,
MonoidT, ⊗,
DistributiveT, ⊗, ⊕,
ZeroAnnihilationT, ⊗ }
end
A semiring is a set of values that has two binary operators ⊕ and ⊗ , such that (a) the values form a commutative
monoid with ⊕ ; (b) the values form a monoid with ⊗ ; (c) ⊗ is distributive over ⊕ ; and (d) ⊗ obeys zero
annihilation, where the zero for ⊗ is the same as the identity for ⊕ .
trait ApproximateRing T extends ApproximateRingT, ⊕, ⊖, ⊗, ≈,
opr ⊕,opr ⊖,opr ⊗,opr ≈
extends { ApproximateAbelianGroupT, ⊕, ⊖, ≈,
ApproximateSemiRingT, ⊕, ⊗, ≈ }
end
An approximate ring is an approximate semiring that also has a unary operator ⊖ that returns inverses for the ⊕
operator so that the values form an approximate group with ⊕ and ⊖ .
trait RingT extends RingT, ⊕, ⊖, ⊗,opr ⊕,opr ⊖,opr ⊗
extends { ApproximateRingT, ⊕, ⊖, ⊗, =,
AbelianGroupT, ⊕, ⊖,
SemiRingT, ⊕, ⊗ }
end
A ring is a semiring that also has a unary operator ⊖ that returns inverses for the ⊕ operator so that the values form
a group with ⊕ and ⊖ .
trait ApproximateCommutativeRing T extends ApproximateCommutativeRingT, ⊕, ⊖, ⊗, ≈,
opr ⊕,opr ⊖,opr ⊗,opr ≈
extends { ApproximateRingT, ⊕, ⊖, ⊗, ≈,
ApproximatelyCommutativeT, ⊗, ≈ }
end
An approximate commutative ring is an approximate ring for which the binary operator ⊗ is also approximately
commutative.
trait CommutativeRing T extends CommutativeRingT, ⊕, ⊖, ⊗,
opr ⊕,opr ⊖,opr ⊗
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