algebra 0.2.0

Abstract algebra for Rust
Documentation 
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// Copyright 2013-2014 The Algebra Developers.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

use ops::Additive;

use structure::GroupAbelianApprox;
use structure::GroupAbelian;
use structure::RingCommutativeApprox;
use structure::RingCommutative;
use structure::FieldApprox;
use structure::Field;


/// A module with approximate operators.
pub trait ModuleApprox<S: RingCommutativeApprox>
    : GroupAbelianApprox<Additive>
{}


/// A module combines two sets: one with an additive abelian group structure and another with a
/// commutative ring structure.
///
/// In addition, and external multiplicative law `∘` is defined. Let `S` be the ring with
/// multiplicative operator noted `×` and multiplicative identity element noted `1`. Then:
///
/// ```notrust
/// ∀ a, b ∈ S
/// ∀ x, y ∈ Self
///
/// a ∘ (x + y) = (a ∘ x) + (a ∘ y)
/// (a + b) ∘ x = (a ∘ x) + (b ∘ x)
/// (a × b) ∘ x = a ∘ (b ∘ x)
/// 1 ∘ x       = x
/// ```
pub trait Module<S: RingCommutative>
    : ModuleApprox<S>
    + GroupAbelian<Additive>
{}

/// A approximate vector space has an approx. module structure over an approx. field.
pub trait VectorSpaceApprox<S: FieldApprox>
    : ModuleApprox<S>
{}


/// A vector space has a module structure over a field instead of a ring.
pub trait VectorSpace<S: Field>
    : VectorSpaceApprox<S>
    + Module<S>
{}