algebra

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.

   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)).

   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.

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