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# algebra

#### package algebra

Package object making available some of the objects in sub-packages

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### Type Members

1. #### trait Abelian extends Semigroup

Abelian/commutative semigroups

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

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.