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// Copyright (C) 2020 Miklos Maroti
// Licensed under the MIT license (see LICENSE)
//! A module that contains the basic sets on which various algebraic structures
//! are implemented.
use crate::*;
use std::fmt::Debug;
/// An arbitrary set of elements where not all representable objects are
/// members of the set. The same element can be represented by different
/// objects, thus the `equals` method shall be used in place of `==`.
pub trait Domain: Clone {
/// The type of the elements of this domain.
type Elem: Clone + Debug;
/// Checks if the given object is a member of the domain.
fn contains(&self, _elem: &Self::Elem) -> bool {
true
}
/// Checks if the given objects represent the same element of the set.
fn equals(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool;
}
/// An arbitrary (multiplicative) semigroup, which is set with an associative binary operation.
pub trait Semigroup: Domain {
/// The multiplicative product of the given elements.
fn mul(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem;
/// The first element is multiplied with the second one in place.
fn mul_assign(&self, elem1: &mut Self::Elem, elem2: &Self::Elem) {
*elem1 = self.mul(elem1, elem2);
}
/// Squares the given element in place.
fn square(&self, elem: &mut Self::Elem) {
*elem = self.mul(elem, elem);
}
}
/// An arbitrary (multiplicative) monoid, which is a semigroup with an identity element.
/// Typical examples are the multiplicative monoid of unitary rings.
pub trait Monoid: Semigroup {
/// The multiplicative identity element of the ring.
fn one(&self) -> Self::Elem;
/// Checks if the given element is the multiplicative identity.
fn is_one(&self, elem: &Self::Elem) -> bool {
self.equals(elem, &self.one())
}
/// Calculates the multiplicative inverse of the given element if it exists.
fn try_inv(&self, elem: &Self::Elem) -> Option<Self::Elem>;
/// Returns true if the given element has a multiplicative inverse.
fn invertible(&self, elem: &Self::Elem) -> bool {
self.try_inv(elem).is_some()
}
}
/// An arbitrary (multiplicative) group, which is a monoid where every element has an inverse.
/// Typical examples are the multiplicative and additive group of rings.
pub trait Group: Monoid {
/// Returns the inverse of the given element element.
fn inv(&self, elem: &Self::Elem) -> Self::Elem {
self.try_inv(elem).unwrap()
}
/// Returns the power of the given element using the square and multiply method.
fn power(&self, mut num: isize, elem: &Self::Elem) -> Self::Elem {
let mut elem = if num < 0 {
num = -num;
self.inv(elem)
} else {
elem.clone()
};
let mut res = self.one();
while num > 0 {
if num % 2 != 0 {
self.mul_assign(&mut res, &elem);
}
num /= 2;
self.square(&mut elem);
}
res
}
}
/// A commutative group written in additive notation. Typical examples
/// are the additive structures of rings, fields and vector spaces.
pub trait AbelianGroup: Domain {
/// The additive identity element of the ring.
fn zero(&self) -> Self::Elem;
/// Checks if the given element is the additive identity of the ring.
fn is_zero(&self, elem: &Self::Elem) -> bool {
self.equals(elem, &self.zero())
}
/// The additive inverse of the given element.
fn neg(&self, elem: &Self::Elem) -> Self::Elem;
/// The element is changed to its additive inverse.
fn neg_assign(&self, elem: &mut Self::Elem) {
*elem = self.neg(elem);
}
/// The additive sum of the given elements
fn add(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem;
/// The second element is added to the first one.
fn add_assign(&self, elem1: &mut Self::Elem, elem2: &Self::Elem) {
*elem1 = self.add(elem1, elem2);
}
/// Doubles the given element in place.
fn double(&self, elem: &mut Self::Elem) {
*elem = self.add(elem, elem);
}
/// The difference of the given elements.
fn sub(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
let mut elem = self.neg(elem2);
self.add_assign(&mut elem, elem1);
elem
}
/// The second element is subtracted from the first one.
fn sub_assign(&self, elem1: &mut Self::Elem, elem2: &Self::Elem) {
self.add_assign(elem1, &self.neg(elem2));
}
/// Returns an integer multiple of the given element.
fn times(&self, num: isize, elem: &Self::Elem) -> Self::Elem {
let group = AdditiveGroup::new(self.clone());
group.power(num, elem)
}
}
/// A ring with an identity element (not necessarily commutative). Typical
/// examples are the rings of rectangular matrices, integers and polynomials.
pub trait UnitaryRing: AbelianGroup + Monoid {
/// Returns the integer multiple of the one element in the ring.
fn int(&self, elem: isize) -> Self::Elem {
self.times(elem, &self.one())
}
}
/// An integral domain is a commutative unitary ring in which the product of
/// non-zero elements are non-zero. This trait not add any new operations,
/// just marks the properties of the ring. A typical examples are the
/// integers, and the ring of polynomials with integer coefficients, which is
/// not an Euclidean domain.
pub trait IntegralDomain: UnitaryRing {
/// Checks if the second element divides the first one and returns the
/// unique quotient if it exists. This method panics if the second element
/// is zero (because there would be no unique solution).
fn try_div(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Option<Self::Elem>;
/// Checks if the first element is a multiple of (or divisible by) the
/// second one.
fn divisible(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool {
if self.is_zero(elem2) {
self.is_zero(elem1)
} else {
self.try_div(elem1, elem2).is_some()
}
}
/// Returns true if the two elements are associates (divide each other)
fn associates(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool {
self.divisible(elem1, elem2) && self.divisible(elem2, elem1)
}
/// We assume, that among all associates of the given elem there is a
/// well defined unique one (non-negative for integers, zero or monic
/// for polynomials). This method returns that representative element.
fn associate_repr(&self, elem: &Self::Elem) -> Self::Elem;
/// Returns the unique invertible element which is the quotient of the
/// associate representative and the given element. This method panics
/// if the element is zero.
fn associate_coef(&self, elem: &Self::Elem) -> Self::Elem;
}
/// An Euclidean domain is an integral domain where the Euclidean algorithm
/// can be implemented. Typical examples are the rings of integers and
/// polynomials.
pub trait EuclideanDomain: IntegralDomain {
/// Performs the euclidean division algorithm dividing the first elem
/// with the second one and returning the quotient and the remainder.
/// This method panics if the second element is zero (because there is
/// no unique solution)
fn quo_rem(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> (Self::Elem, Self::Elem);
/// Performs the division just like the [quo_rem](EuclideanDomain::quo_rem)
/// method would do and returns the quotient. This method panics if the
/// second element is zero.
fn quo(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
self.quo_rem(elem1, elem2).0
}
/// Performs the division just like the [quo_rem](EuclideanDomain::quo_rem)
/// method would do and returns the remainder. This method panics if the
/// second element is zero.
fn rem(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
self.quo_rem(elem1, elem2).1
}
/// Returns true if the second element is zero or the first element
/// has zero quotient by the second one. These are the representative
/// elements of the factor ring by the principal ideal generated by
/// the second element.
fn reduced(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool {
self.is_zero(elem2) || self.is_zero(&self.quo(elem1, elem2))
}
/// Calculates the greatest common divisor of two elements using the
/// Euclidean algorithm.
fn gcd(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
let mut elem1 = elem1.clone();
let mut elem2 = elem2.clone();
while !self.is_zero(&elem2) {
let rem = self.rem(&elem1, &elem2);
elem1 = elem2;
elem2 = rem;
}
elem1
}
/// Calculates the lest common divisor of the two elements.
fn lcm(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
let gcd = self.gcd(elem1, elem2);
if self.is_zero(&gcd) {
gcd
} else {
self.mul(&self.quo(elem1, &gcd), elem2)
}
}
/// Performs the extended Euclidean algorithm which returns the greatest
/// common divisor, and two elements that multiplied with the inputs gives
/// the greatest common divisor.
fn extended_gcd(
&self,
elem1: &Self::Elem,
elem2: &Self::Elem,
) -> (Self::Elem, Self::Elem, Self::Elem) {
if self.is_zero(elem2) {
(elem1.clone(), self.one(), self.zero())
} else {
let (quo, rem) = self.quo_rem(elem1, elem2);
let (gcd, x, y) = self.extended_gcd(elem2, &rem);
let z = self.sub(&x, &self.mul(&y, &quo));
(gcd, y, z)
}
}
/// Checks if the given two elements are relative prime.
fn relative_primes(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool {
let gcd = self.gcd(elem1, elem2);
self.invertible(&gcd)
}
}
/// A field is a commutative ring with identity where each non-zero element
/// has a multiplicative inverse. Typical examples are the real, complex and
/// rational numbers, and finite fields constructed from an Euclidean domain
/// and one of its irreducible elements. All fields are Euclidean domains
/// themselves, with a rather trivial structure.
pub trait Field: EuclideanDomain {
/// Returns the multiplicative inverse of the given non-zero element.
/// This method panics for the zero element.
fn inv(&self, elem: &Self::Elem) -> Self::Elem;
/// Returns the quotient of the two elements in the field. This method
/// panics if the second element is zero.
fn div(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
self.mul(elem1, &self.inv(elem2))
}
/// Returns an integer power of the given element.
fn power(&self, num: isize, elem: &Self::Elem) -> Self::Elem {
if self.is_zero(elem) {
self.zero()
} else {
let group = MultiplicativeGroup::new(self.clone());
group.power(num, elem)
}
}
}
impl<A: Field> IntegralDomain for A {
fn try_div(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Option<Self::Elem> {
Some(self.div(elem1, elem2))
}
fn associate_repr(&self, elem: &Self::Elem) -> Self::Elem {
if self.is_zero(elem) {
self.zero()
} else {
self.one()
}
}
fn associate_coef(&self, elem: &Self::Elem) -> Self::Elem {
self.inv(elem)
}
}
impl<A: Field> EuclideanDomain for A {
fn quo_rem(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> (Self::Elem, Self::Elem) {
assert!(!self.is_zero(elem2));
(self.div(elem1, elem2), self.zero())
}
fn quo(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
assert!(!self.is_zero(elem2));
self.div(elem1, elem2)
}
fn rem(&self, _elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
assert!(!self.is_zero(elem2));
self.zero()
}
fn reduced(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool {
self.is_zero(elem1) || self.is_zero(elem2)
}
fn gcd(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
if self.is_zero(elem1) && self.is_zero(elem2) {
self.zero()
} else {
self.one()
}
}
fn lcm(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem {
if self.is_zero(elem1) || self.is_zero(elem2) {
self.zero()
} else {
self.one()
}
}
}
/// A set with a reflexive, transitive and antisymmetric relation.
/// Typical examples are the lattices and the divisibility relation of
/// integral domains (which might not be a lattice).
pub trait PartialOrder: Domain {
/// Returns true if the first element is less than or equal to the
/// second one in the partial order.
fn leq(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool;
/// Returns true if the first element is strictly less than the
/// second one.
fn less_than(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool {
!self.equals(elem1, elem2) && self.leq(elem1, elem2)
}
/// Returns true if one of the elements is less than or equal to
/// the other.
fn comparable(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> bool {
self.leq(elem1, elem2) || self.leq(elem2, elem1)
}
}
/// A partial order that has a largest and smallest element. Typical
/// examples are bounded lattices.
pub trait BoundedOrder: PartialOrder {
/// Returns the largest element of the lattice.
fn max(&self) -> Self::Elem;
/// Returns the smallest element of the lattice.
fn min(&self) -> Self::Elem;
}
/// A set where the join and meet of elements can be calculated. Typical
/// examples are the total orders of integers or the divisibility relation
/// on the associate classes of an Euclidean domain.
pub trait Lattice: PartialOrder {
/// Returns the largest element that is less than or equal to both given
/// elements.
fn meet(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem;
/// Returns the smallest element that is greater than or equal to both
/// given elements.
fn join(&self, elem1: &Self::Elem, elem2: &Self::Elem) -> Self::Elem;
}
/// A lattice that is distributive. No new methods are added.
pub trait DistributiveLattice: Lattice {}
/// A Boolean algebra, which is a complemented bounded distributive lattice.
/// Typical examples are the two-element boolean algebra and the power sets.
pub trait BooleanAlgebra: DistributiveLattice + BoundedOrder {
/// Returns the complement (logical negation) of the given element.
fn not(&self, elem: &Self::Elem) -> Self::Elem;
}