Interface Semiring<T extends Semiring<T>>

Type Parameters:
T - the type of the semiring element.
All Superinterfaces:
Comparable<T>, Serializable
All Known Subinterfaces:
Field<T>, Ring<T>
All Known Implementing Classes:
BoolSemiring, Complex128, Complex64, RealFloat32, RealFloat64, RealInt16, RealInt32
public interface Semiring<T extends Semiring<T>> extends Comparable<T>, Serializable
Defines a mathematical semiring structure and specifies the operations that semiring elements must support.

A semiring is an algebraic structure consisting of a set R equipped with two binary operations: addition (+) and multiplication (*). The semiring generalizes the concept of a ring but does not require the existence of additive inverses (i.e., negative elements). As a result, subtraction is not generally defined within a semiring.

Formal Definition:

For all elements a, b, and c in R, the following properties must hold:

  • Addition is associative: a + (b + c) = (a + b) + c
  • Addition is commutative: a + b = b + a
  • Additive identity exists: There exists an element 0 in R such that a + 0 = a
  • Multiplication is associative: a * (b * c) = (a * b) * c
  • Multiplicative identity exists: There exists an element 1 in R such that a * 1 = a
  • Multiplication distributes over addition:
    • Left distributivity: a * (b + c) = (a * b) + (a * c)
    • Right distributivity: (a + b) * c = (a * c) + (b * c)
  • Zero is absorbing for multiplication: a * 0 = 0 * a = 0

Implementations:

Implementations of the Semiring interface should ensure that instances are immutable. This means that all operations should return new instances rather than modifying existing ones. Immutability guarantees thread safety and consistent behavior across different contexts.

The compareTo(Semiring) method should implement some ordering (total or partial) on the semiring.

Further, implementations should ensure that the equality and hash code methods are consistent with the semiring's equality definition.

Examples of Semirings:

  • Natural Numbers (ℕ): The set of natural numbers with usual addition and multiplication forms a semiring.
  • Boolean Semiring: The set {false, true} with logical OR as addition and logical AND as multiplication.
  • Tropical Semiring: The set of real numbers extended with infinity, where addition is defined as taking the minimum (or maximum) and multiplication as the standard real number addition.

Interface Methods:

The Semiring interface specifies the following methods that semiring elements must implement:

  • add(Semiring): Performs the addition operation, returning a new semiring element.
  • mult(Semiring): Performs the multiplication operation, returning a new semiring element.
  • isZero(): Checks if the element is the additive identity (zero element).
  • isOne(): Checks if the element is the multiplicative identity (one element).
  • getZero(): Returns the additive identity element of the semiring.
  • getOne(): Returns the multiplicative identity element of the semiring.

Usage Example:


 // Assume T is a concrete implementation of Semiring<T>
 T a = ...; // Initialize element a
 T b = ...; // Initialize element b

 T sum = a.add(b);       // Perform addition in the semiring
 T product = a.mult(b);  // Perform multiplication in the semiring

 boolean isZero = a.isZero(); // Check if a is the additive identity
 boolean isOne = b.isOne();   // Check if b is the multiplicative identity
See Also:

  • Method Summary

    Modifier and Type
    Method
    Description
    T
    add(T b)
    Sums two elements of this semiring (associative and commutative).
    int
    compareTo(T b)
    Compares this element of the semiring with b.
    double
    doubleValue()
    Converts this semiring value to an equivalent double value.
    T
    getOne()
    Gets the multiplicative identity for this semiring.
    T
    getZero()
    Gets the additive identity for this semiring.
    boolean
    isOne()
    Checks if this value is a multiplicative identity for this semiring.
    boolean
    isZero()
    Checks if this value is an additive identity for this semiring.
    T
    mult(T b)
    Multiplies two elements of this semiring (associative).

  • Method Details

    • add

      T add(T b)
      Sums two elements of this semiring (associative and commutative).
      Parameters:
      b - Second semiring element in sum.
      Returns:
      The sum of this element and b.

    • mult

      T mult(T b)
      Multiplies two elements of this semiring (associative).
      Parameters:
      b - Second semiring element in product.
      Returns:
      The product of this semiring element and b.

    • isZero

      boolean isZero()

      Checks if this value is an additive identity for this semiring.

      An element 0 is an additive identity if a + 0 = a for any a in the semiring.

      Returns:
      True if this value is an additive identity for this semiring. Otherwise, false.

    • isOne

      boolean isOne()

      Checks if this value is a multiplicative identity for this semiring.

      An element 1 is a multiplicative identity if a * 1 = a for any a in the semiring.

      Returns:
      True if this value is a multiplicative identity for this semiring. Otherwise, false.

    • getZero

      T getZero()

      Gets the additive identity for this semiring.

      An element 0 is an additive identity if a + 0 = a for any a in the semiring.

      Returns:
      The additive identity for this semiring.

    • getOne

      T getOne()

      Gets the multiplicative identity for this semiring.

      An element 1 is a multiplicative identity if a * 1 = a for any a in the semiring.

      Returns:
      The multiplicative identity for this semiring.

    • compareTo

      int compareTo(T b)
      Compares this element of the semiring with b.
      Specified by:
      compareTo in interface Comparable<T extends Semiring<T>>
      Parameters:
      b - Second element of the semiring.
      Returns:
      An int value:
      • 0 if this semiring element is equal to b.
      • invalid input: '<' 0 if this semiring element is less than b.
      • > 0 if this semiring element is greater than b.
        • Hence, this method returns zero if and only if the two semiring elements are equal, a negative value if and only the semiring element it was called on is less than b and positive if and only if the semiring element it was called on is greater than b.

    • doubleValue

      double doubleValue()
      Converts this semiring value to an equivalent double value.
      Returns:
      A double value equivalent to this semiring element.