Module structure

Fundamental algebraic structures.

|(• ◡•)|ノ〵(❍ᴥ❍⋃)     - "ALGEBRAIC!!!"

For most applications requiring an abstraction over the reals, FieldApprox should be sufficient.

Fundamental algebraic structures

Most of these traits also come in an approximate flavor for types that do not satisfy the required properties exactly, but would still benefit from abstractions over the structure in question.

Algebraic properties

The goal of algebraic structures is to allow elements of sets to be combined together using one or several operators. The number and properties of those operators characterize the algebraic structure. Abstract operators are usually noted ∘, +, or ×. The last two are preferred when their behavior conform with the usual meaning of addition and multiplication of reals. Let Self be a set. Here is a list of the most common properties those operator may fulfill:

(Closure)       a, b ∈ Self ⇒ a ∘ b ∈ Self, 
(Divisibility)  ∀ a, b ∈ Self, ∃! r, l ∈ Self such that l ∘ a = b and a ∘ r = b
(Invertibility) ∃ e ∈ Self, ∀ a ∈ Self, ∃ r, l ∈ Self such that l ∘ a = a ∘ r = e
                If the right and left inverse are equal they are usually noted r = l = a⁻¹.
(Associativity) ∀ a, b, c ∈ Self, (a ∘ b) ∘ c = a ∘ (b ∘ c)       
(Neutral Elt.)  ∃ e ∈ Self, ∀ a ∈ Self, e ∘ a = a ∘ e = a
(Commutativity) ∀ a, b ∈ Self, a ∘ b = b ∘ a

When one works with inexact arithmetic, e.g. using floating point numbers, those properties cannot possibly be satisfied due to the discrete nature of our computing tools. Thus a looser, approximate, version is available. Note that fulfilling a property listed above implies that its approximate version is fulfilled as well.

(Approx. Closure) a, b ∈ Self ⇒ ∃ c ≈ a ∘ b such that c ∈ Self, 
(Approx. Div.)    ∀ a, b ∈ Self, ∃ r, l ∈ Self such that l ∘ a ≈ b and a ∘ r ≈ b
(Approx. Inv.)    ∃ e ∈ Self, ∀ a ∈ Self, ∃ r, l ∈ Self such that l ∘ a ≈ e and a ∘ r ≈ e
(Approx. Assoc.)  ∀ a, b, c ∈ Self, (a ∘ b) ∘ c ≈ a ∘ (b ∘ c)       
(Approx. Neutr.)  ∃ e ∈ Self, ∀ a ∈ Self, e ∘ a ≈ a and a ∘ e ≈ a
(Approx. Commut.) ∀ a, b ∈ Self, a ∘ b ≈ b ∘ a

Identity elements

Two traits are provided that allow the definition of the additive and multiplicative identity elements:

• IdentityAdditive
• IdentityMultiplicative

Group-like structures

These structures are provided for both the addition and multiplication.

              Magma
                |
        _______/ \______
       /                \
 divisibility      associativity
      |                  |
      V                  V
 Quasigroup          Semigroup
      |                  |
  identity            identity
      |                  |
      V                  V
    Loop               Monoid
      |                  |
 associativity     invertibility
       \______   _______/
              \ /
               |
               V
             Group
               |
         commutativity
               |
               V
          AbelianGroup

The following traits are provided:

•  `Closure`(`Additive`|`Multiplicative`)(`Approx`)?

•    `Magma`(`Additive`|`Multiplicative`)(`Approx`)?

• Quasigroup(Additive|Multiplicative)(Approx)?

•     `Loop`(`Additive`|`Multiplicative`)(`Approx`)?

• Semigroup(Additive|Multiplicative)(Approx)?

•   `Monoid`(`Additive`|`Multiplicative`)(`Approx`)?

•    `Group`(`Additive`|`Multiplicative`)(`Approx`)?

• AbelianGroup(Additive|Multiplicative)(Approx)?

Ring-like structures

GroupAdditiveAbelian     MonoidMultiplicative
          \________   ________/
                   \ /
                    |
                    V
                   Ring
                    |
           commutativity_of_mul
                    |
                    V
             RingCommutative     GroupMultiplicativeAbelian
                     \_______   ___________/
                             \ /
                              |
                              V
                            Field

The following traits are provided:

•        `Ring`(`Approx`)?

• RingCommutative(Approx)?

•       `Field`(`Approx`)?

Module-like structures

GroupAdditiveAbelian     RingCommutative
          \______         _____/
                 \       /
                  |     |
                  V     V
               Module<Scalar>          Field
                   \______         _____/
                          \       /
                           |     |
                           V     V
                        VectorSpace<Scalar>

The following traits are provided:

• Module(Approx)?
• VectorSpace(Approx)?

Quickcheck properties

Functions are provided to test that algebraic properties like assciociativity and commutativity hold for a given set of arguments.

For example:

#[quickcheck]
fn prop_mul_is_associative_approx(args: (i32, i32, i32)) -> bool {
    SemigroupMultiplicativeApprox::prop_mul_is_associative_approx(args)
}

Traits

Field

A field is a commutative ring, and an abelian group under the multiplication operator.

FieldApprox

An approximate field is an approx. commutative ring, and an approx. abelian group under the multiplication operator.

Group

A group is a loop and a monoid at the same time.

GroupAbelian

A commutative group.

GroupAbelianApprox

An approximately commutative group.

GroupApprox

An approximate group is an approx. loop and an approx. monoid simultaneously.

Loop

A quasigroup with an unique identity element.

LoopApprox

An approximate quasigroup with an unique identity element.

Magma

Types that are closed under a given operator.

MagmaApprox

Types that are approximately closed under a given operator.

Module

A module combines two sets: one with an additive abelian group structure and another with a commutative ring structure.

ModuleApprox

A module with approximate operators.

Monoid

A semigroup equipped with an identity element.

MonoidApprox

An approximate semigroup equipped with an identity element.

Quasigroup

A magma with the divisibility property.

QuasigroupApprox

A magma with the approximate divisibility property.

Ring

A ring is the combination of an abelian group and a multiplicative monoid structure.

RingApprox

An approximate ring is given the approximate version of the ring’s properties.

RingCommutative

An ring with a commutative multiplication.

RingCommutativeApprox

An approximative ring with an approximately commutative multiplication.

Semigroup

An associative magma.

SemigroupApprox

An approximately associative magma.

VectorSpace

A vector space has a module structure over a field instead of a ring.

VectorSpaceApprox

A approximate vector space has an approx. module structure over an approx. field.