Package

ap

algebra

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package algebra

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

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Type Members

  1. trait Abelian extends Semigroup

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    Abelian/commutative semigroups

  2. trait CommutativePseudoRing extends PseudoRing

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  3. trait CommutativeRing extends Ring with CommutativePseudoRing

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  4. trait EuclidianRing extends CommutativeRing with RingWithDivision

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    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

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

  6. trait Group extends Monoid

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    Groups are monoids that additionally have inverses

  7. trait Monoid extends Semigroup

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    Monoids are semigroups with a neutral element (or zero)

  8. trait OrderedRing extends Ring with RingWithOrder

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    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

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    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

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    Rings are structures with both addition and multiplication

  11. trait RingWithDivision extends PseudoRing

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

  12. trait RingWithIntConversions extends PseudoRing

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    Ring that can also convert ring elements back to integers.

  13. trait RingWithOrder extends PseudoRing

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    Rings that also possess an ordering relation

  14. trait Semigroup extends AnyRef

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Value Members

  1. val IntegerGroup: Group with Abelian

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    The additive group of integers

  2. object IntegerRing extends EuclidianRing with OrderedRing with CommutativeRing

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    The built-in ring of integers

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