org.tensorics.core.math.structures.ringlike

Interface Semiring<T>

Type Parameters:

T - the type of the elements of the structure

All Superinterfaces:

RinglikeStructure<T>, Structure<T>

All Known Subinterfaces:

Field<T>, OrderedField<T>, Ring<T>

All Known Implementing Classes:

AbstractOrderedField, DoubleField

public interface Semiring<T>
extends RinglikeStructure<T>

The algebraic structure of a semiring, which has the following properties:

R is the underlying set; a, b, c are elements of R.

1. (R, +) is a commutative monoid with identity element 0:

   (a + b) + c = a + (b + c)

   0 + a = a + 0 = a

   a + b = b + a

2. (R, *) is a monoid with identity element 1:

   (a*b)*c = a*(b*c)

   1*a = a*1 = a

3. Multiplication left and right distributes over addition:

   a*(b + c) = (a*b) + (a*c) = (a + b)*c = (a*c) + (b*c)

4. Multiplication by 0 annihilates R:

Method Summary

Modifier and Type	Method and Description
CommutativeMonoid<T>	additionStructure() Returns the grouplike structure representing the addition of elements.
Monoid<T>	multiplicationStructure() Returns the grouplike structure representing the multiplication of elements.

Method Detail

additionStructure

CommutativeMonoid<T> additionStructure()

Description copied from interface: RinglikeStructure

Returns the grouplike structure representing the addition of elements.

Specified by:

additionStructure in interface RinglikeStructure<T>

Returns:

the group like structure for the addition

multiplicationStructure

Monoid<T> multiplicationStructure()

Description copied from interface: RinglikeStructure

Returns the grouplike structure representing the multiplication of elements.

Specified by:

multiplicationStructure in interface RinglikeStructure<T>

Returns:

the group like structure for multiplication