Module general

Fundamental algebraic structures.

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

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

Identity elements

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

• IdentityAdditive
• IdentityMultiplicative

AbstractGroup-like structures

These structures are provided for both the addition and multiplication.

These can be derived automatically by alga_traits attribute from alga_derive crate.

           AbstractMagma
                |
        _______/ \______
       /                \
 divisibility       associativity
      |                  |
      V                  V
AbstractQuasigroup AbstractSemigroup
      |                  |
  identity            identity
      |                  |
      V                  V
 AbstractLoop       AbstractMonoid
      |                  |
 associativity     invertibility
       \______   _______/
              \ /
               |
               V
         AbstractGroup
               |
         commutativity
               |
               V
     AbstractGroupAbelian

The following traits are provided:

• (Abstract|Additive|Multiplicative)Magma
• (Abstract|Additive|Multiplicative)Quasigroup
• (Abstract|Additive|Multiplicative)Loop
• (Abstract|Additive|Multiplicative)Semigroup
• (Abstract|Additive|Multiplicative)Monoid
• (Abstract|Additive|Multiplicative)Group
• (Abstract|Additive|Multiplicative)GroupAbelian

Ring-like structures

These can be derived automatically by alga_traits attribute from alga_derive crate.

     GroupAbelian           Monoid
          \________   ________/
                   \ /
                    |
                    V
                   Ring
                    |
           commutativity_of_mul
                    |
                    V
             RingCommutative           GroupAbelian
                     \_______   ___________/
                             \ /
                              |
                              V
                            Field

The following traits are provided:

• Ring
• RingCommutative
• Field

Module-like structures

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

The following traits are provided:

• Module
• VectorSpace

Quickcheck properties

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

These tests can be automatically derived by alga_quickcheck attribute from alga_derive crate.

For example:

use algebra::general::SemigroupMultiplicative;

quickcheck! {
    fn prop_mul_is_associative(args: (i32, i32, i32)) -> bool {
        SemigroupMultiplicative::prop_mul_is_associative(args)
    }
}

Structs

• Additive
  The addition operator, commonly symbolized by +.
• Id
  The universal identity element wrt. a given operator, usually noted Id with a context-dependent subscript.
• Multiplicative
  The multiplication operator, commonly symbolized by ×.

Traits

• AbstractField
  A field is a commutative ring, and an Abelian group under both operators.
• AbstractGroup
  A group is a loop and a monoid at the same time.
• AbstractGroupAbelian
  An Abelian group is a commutative group.
• AbstractLoop
  A loop is a quasigroup with an unique identity element, e.
• AbstractMagma
  A magma is an algebraic structure which consists of a set equipped with a binary operation, ∘, which must be closed.
• AbstractModule
  A module combines two sets: one with an Abelian group structure and another with a commutative ring structure.
• AbstractMonoid
  A monoid is a semigroup equipped with an identity element, e.
• AbstractQuasigroup
  A quasigroup is a magma which that has the divisibility property (or Latin square property). A set with a closed binary operation with the divisibility property.
• AbstractRing
  A ring is the combination of an Abelian group and a multiplicative monoid structure.
• AbstractRingCommutative
  A ring with a commutative multiplication.
• AbstractSemigroup
  A semigroup is a quasigroup that is associative.
• AdditiveGroup
  [Alias] Algebraic structure specialized for one kind of operation.
• AdditiveGroupAbelian
  [Alias] Algebraic structure specialized for one kind of operation.
• AdditiveLoop
  [Alias] Algebraic structure specialized for one kind of operation.
• AdditiveMagma
  [Alias] Algebraic structure specialized for one kind of operation.
• AdditiveMonoid
  [Alias] Algebraic structure specialized for one kind of operation.
• AdditiveQuasigroup
  [Alias] Algebraic structure specialized for one kind of operation.
• AdditiveSemigroup
  [Alias] Algebraic structure specialized for one kind of operation.
• ClosedAdd
  [Alias] Trait alias for Add and AddAssign with result of type Self.
• ClosedDiv
  [Alias] Trait alias for Div and DivAssign with result of type Self.
• ClosedMul
  [Alias] Trait alias for Mul and MulAssign with result of type Self.
• ClosedNeg
  [Alias] Trait alias for Neg with result of type Self.
• ClosedSub
  [Alias] Trait alias for Sub and SubAssign with result of type Self.
• ComplexField
  Trait shared by all complex fields and its subfields (like real numbers).
• Field
  [Alias] Algebraic structure specialized for one kind of operation.
• Identity
  A type that is equipped with identity.
• JoinSemilattice
  A set where every two elements have a supremum (i.e. smallest upper bound).
• Lattice
  Partially orderable sets where every two elements have a supremum and infimum.
• MeetSemilattice
  A set where every two elements have an infimum (i.e. greatest lower bound).
• Module
  A module which overloads the * and + operators.
• MultiplicativeGroup
  [Alias] Algebraic structure specialized for one kind of operation.
• MultiplicativeGroupAbelian
  [Alias] Algebraic structure specialized for one kind of operation.
• MultiplicativeLoop
  [Alias] Algebraic structure specialized for one kind of operation.
• MultiplicativeMagma
  [Alias] Algebraic structure specialized for one kind of operation.
• MultiplicativeMonoid
  [Alias] Algebraic structure specialized for one kind of operation.
• MultiplicativeQuasigroup
  [Alias] Algebraic structure specialized for one kind of operation.
• MultiplicativeSemigroup
  [Alias] Algebraic structure specialized for one kind of operation.
• Operator
  Trait implemented by types representing abstract operators.
• RealDeprecated
  The field of reals. This has been renamed to RealField.
• RealField
  Trait shared by all reals.
• Ring
  [Alias] Algebraic structure specialized for one kind of operation.
• RingCommutative
  [Alias] Algebraic structure specialized for one kind of operation.
• SubsetOf
  Nested sets and conversions between them (using an injective mapping). Useful to work with substructures. In generic code, it is preferable to use SupersetOf as trait bound whenever possible instead of SubsetOf (because SupersetOf is automatically implemented whenever SubsetOf is).
• SupersetOf
  Nested sets and conversions between them. Useful to work with substructures. It is preferable to implement the SupersetOf trait instead of SubsetOf whenever possible (because SupersetOf is automatically implemented whenever SubsetOf is.
• TwoSidedInverse
  Trait used to define the two_sided_inverse element relative to the given operator.