Type Members

1.  trait Abelian extends Semigroup

   Abelian/commutative semigroups

2.  trait CommutativePseudoRing extends PseudoRing
3.  trait CommutativeRing extends Ring with CommutativePseudoRing
4.  trait EuclidianRing extends CommutativeRing with RingWithDivision

   Euclidian rings extend rings with operations for division and remainder, with the Euclidian definition: plus(mul(div(s, t), t), mod(s, t)) === s, with f(mod(s, t)) in [0, abs(t)) for some appropriate embedding into real numbers.

5.  trait Field extends CommutativeRing with RingWithDivision

   Fields are commutative rings in which all non-zero elements have multiplicative inverses.

6.  trait Group extends Monoid

   Groups are monoids that additionally have inverses

7.  trait Monoid extends Semigroup

   Monoids are semigroups with a neutral element (or zero)

8.  trait OrderedRing extends Ring with RingWithOrder

   Ordered rings are rings with ordering relation in which addition, multiplication, and ordering are consistent: leq(s, t) ==> leq(plus(s, a), plus(t, a)) and leq(zero, s) & leq(zero, t) ==> leq(zero, mul(s, t)).

9.  trait PseudoRing extends AnyRef

   A Pseudo-ring is a structure with the same operations as a ring, but no guarantee that multiplication satisfies the ring axioms

10.  trait Ring extends PseudoRing

    Rings are structures with both addition and multiplication

11.  trait RingWithDivision extends PseudoRing

    Rings that also have a division operation (though possibly not satisfying the standard axioms)

12.  trait RingWithIntConversions extends PseudoRing

    Ring that can also convert ring elements back to integers.

13.  trait RingWithOrder extends PseudoRing

    Rings that also possess an ordering relation

14.  trait Semigroup extends AnyRef
15.  trait SymbolicTimes extends Semigroup

    Semigroups that provide a symbolic times operator, which accepts terms as both arguments.