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package poly.algebra
import poly.algebra.factory._
import poly.algebra.specgroup._
/**
* Represents a semiring. A semiring is a commutative additive monoid as well as a multiplicative monoid.
*
* An instance of this typeclass should satisfy the following axioms:
*
*
$lawAdditiveAssociativity
*
$lawAdditiveIdentity
*
$lawAdditiveCommutativity
*
$lawMultiplicativeAssociativity
*
$lawMultiplicativeIdentity
*
$lawDistributivityMA
*
$lawAnnihilationM
*
* @define lawDistributivityMA '''Distributivity of multiplication w.r.t. addition''': ∀''a'', ''b'', ''c''∈X, (''a'' + ''b'') * ''c'' == ''a'' * ''c'' + ''b'' * ''c'' and ''a'' * (''b'' + ''c'') == ''a'' * ''b'' + ''a'' * ''c''.
* @define lawAnnihilationM '''Zero as an annihilator''': ∀''a''∈X, 0 * ''a'' == ''a'' * 0 == 0.
* @author Tongfei Chen
* @since 0.1.0
*/
trait Semiring[@sp(fdi) X] extends AdditiveCMonoid[X] with MultiplicativeMonoid[X] {
/** Returns the 2 element in this semiring. */
def two = add(one, one)
def product[Y](that: Semiring[Y]): Semiring[(X, Y)] = new SemiringT.Product(this, that)
}
object Semiring extends ImplicitGetter[Semiring] {
/** Creates a semiring using the provided operations `add`, `mul`, `zero`, and `one`. */
def create[@sp(di) X](fAdd: (X, X) => X, fMul: (X, X) => X, zeroElem: X, oneElem: X)
: Semiring[X] = new Semiring[X] {
def add(x: X, y: X): X = fAdd(x, y)
def mul(x: X, y: X): X = fMul(x, y)
def zero: X = zeroElem
def one: X = oneElem
}
def product[X, Y](implicit X: Semiring[X], Y: Semiring[Y]): Semiring[(X, Y)] = new SemiringT.Product(X, Y)
}
private[poly] object SemiringT {
class Product[X, Y](X: Semiring[X], Y: Semiring[Y]) extends Semiring[(X, Y)] {
def add(x: (X, Y), y: (X, Y)) = (X.add(x._1, y._1), Y.add(x._2, y._2))
def one = (X.one, Y.one)
def mul(x: (X, Y), y: (X, Y)) = (X.mul(x._1, y._1), Y.mul(x._2, y._2))
def zero = (X.zero, Y.zero)
}
}
