Trait SemigroupHom

Source

pub trait SemigroupHom {
    type Source: Semigroup;
    type Target: Semigroup;

    // Required method
    fn apply(&self, x: &Self::Source) -> Self::Target;
}

Expand description

A semigroup homomorphism: a structure-preserving map between semigroups.

A homomorphism f: S → T preserves the semigroup operation:

Laws (not enforced by type system):

Preserve combine: f(x.combine(y)) == f(x).combine(f(y))

§Example

use algebra_core::{Semigroup, Monoid, SemigroupHom};

// Wrapper for usize with addition
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
struct Sum(usize);

impl Semigroup for Sum {
    fn combine(&self, other: &Self) -> Self {
        Sum(self.0 + other.0)
    }
}

impl Monoid for Sum {
    fn empty() -> Self { Sum(0) }
}

// String length is a homomorphism from (String, concat) to (Sum, +)
struct Length;

impl SemigroupHom for Length {
    type Source = String;
    type Target = Sum;

    fn apply(&self, s: &String) -> Sum {
        Sum(s.len())
    }
}

// Verify: len(s1 + s2) = len(s1) + len(s2)
let len = Length;
let s1 = String::from("hello");
let s2 = String::from("world");
assert_eq!(
    len.apply(&s1.clone().combine(&s2)),
    len.apply(&s1).combine(&len.apply(&s2))
);