Module maths_traits::algebra::module_like

Traits for structures with an addition and scalar multiplication operation

Implementation

This module builds upon the and behaves in a way similar to the group-like structures, though, at this point in time, there are no additional properties or operations for structs to consider when targeting this module's systems.

Usage

Similarly to the group-like module, there are a number of trait aliases corresponding to a system of module-like algebraic structures that form a heirarchy as represented in the following diagram:

        Affine Space    Additive Abelian Group
             |                   |
             |              Ring Module
             |                   |
             ----Vector Space-----
                      |
                   Algebra

Where:

• A ring-module is an additive abelian group with a scalar multiplication operation with elements from a particular [Ring]
• A vector-space is a ring-module with scalars that form a field
• An algebra is a vector-space with a distributive multiplication operation
• An affine-space is a set with a subtraction operation producing a vector
  • for more information see the trait-level documentation

Traits

Add	The addition operator +.
AddAssign	The addition assignment operator +=.
BilinearForm	A SesquilinearForm where x•(c*y) = (x•y)*c and σ(x) = x
ComplexSesquilinearForm	A SesquilinearForm where x•(c*y) = (x•y)*̅c and σ(x) = ̅x
Div	The division operator /.
DivAssign	The division assignment operator /=.
Index	Used for indexing operations (container[index]) in immutable contexts.
IndexMut	Used for indexing operations (container[index]) in mutable contexts.
Mul	The multiplication operator *.
MulAssign	The multiplication assignment operator *=.
Neg	The unary negation operator -.
ReflexiveForm	A SesquilinearForm where x•y = 0 implies y•x = 0
SesquilinearForm	A product between two vectors or module elements resulting in a scalar that is semi-linear in both arguments
SkewSesquilinearForm	A SesquilinearForm where σ(x•y) = -y•x
Sub	The subtraction operator -.
SubAssign	The subtraction assignment operator -=.
SymSesquilinearForm	A SesquilinearForm where σ(x•y) = y•x

Trait aliases

AffineSpace	A set of points without an origin
Algebra	A vector space with a distributive multiplication operation
AssociativeAlgebra	An algebra with an associative multiplication operation
AssociativeRingAlgebra	A ring algebra with an associative multiplication operation
HermitianForm	A ComplexSesquilinearForm where x•y = ̅y̅•̅x
RingAlgebra	A ring module with a distributive multiplication operation
RingModule	An abelian additive group with a distributive scalar multiplication with a unital ring
SkewHermitianForm	A ComplexSesquilinearForm where x•y = -̅y̅•̅x
SkewSymmetricForm	A BilinearForm where x•y = -y•x
SymmetricForm	A BilinearForm where x•y = y•x
UnitalAlgebra	An algebra with a multiplicative identity
UnitalRingAlgebra	A ring algebra with a multiplicative identity
VectorSpace	An abelian additive group with a distributive scalar multiplication with a field