Class Integers

java.lang.Object
org.episteme.core.mathematics.sets.Integers
All Implemented Interfaces:
AbelianGroup<Integer>, AbelianMonoid<Integer>, Group<Integer>, Magma<Integer>, Monoid<Integer>, Ring<Integer>, Semiring<Integer>, InfiniteSet<Integer>, Set<Integer>
public final class Integers extends Object implements Ring<Integer>, InfiniteSet<Integer>
The structure of integers (ℤ = {..., -2, -1, 0, 1, 2, ...}).

This class represents the structure of integers, not individual elements. It implements Ring because ℤ forms a commutative ring with unity under addition and multiplication.

Integers are an infinite, countable set.

Usage

Integers structure = Integers.getInstance();
Integer five = Integer.of(5);
Integer negThree = Integer.of(-3);

// Use structure for operations
Integer two = structure.add(five, negThree);
Integer negFifteen = structure.multiply(five, negThree);
Integer negFive = structure.negate(five);
Since:
1.0
Author:
Gemini AI (Google DeepMind)
See Also:

  • Method Details

    • getInstance

      public static Integers getInstance()
      Returns the singleton instance.
      Returns:
      the Integers structure

    • operate

      public Integer operate(Integer a, Integer b)
      Description copied from interface: Magma
      Performs the binary operation on two elements.

      This is the fundamental operation of a magma. The result must be an element of this magma (closure property).

      Properties: None required (not necessarily associative or commutative)

      Specified by:
      operate in interface Magma<Integer>
      Parameters:
      a - the first operand
      b - the second operand
      Returns:
      the result of a ∗ b
      See Also:

    • add

      public Integer add(Integer a, Integer b)
      Description copied from interface: AbelianMonoid
      Returns the sum of two elements (additive notation).

      Delegates to Magma.operate(Object, Object).

      Specified by:
      add in interface AbelianMonoid<Integer>
      Parameters:
      a - the first addend
      b - the second addend
      Returns:
      a + b

    • zero

      public Integer zero()
      Description copied from interface: AbelianMonoid
      Returns the additive identity (zero element).

      Delegates to AbelianMonoid.identity().

      Specified by:
      zero in interface AbelianMonoid<Integer>
      Returns:
      the zero element

    • negate

      public Integer negate(Integer element)
      Description copied from interface: AbelianGroup
      Returns the additive inverse (negation) of an element.

      Satisfies: a + (-a) = (-a) + a = 0

      Specified by:
      negate in interface AbelianGroup<Integer>
      Parameters:
      element - the element to negate
      Returns:
      -element
      See Also:

    • multiply

      public Integer multiply(Integer a, Integer b)
      Description copied from interface: Semiring
      Returns the product of two elements.

      Multiplication must be associative and distribute over addition.

      Specified by:
      multiply in interface Semiring<Integer>
      Parameters:
      a - the first factor
      b - the second factor
      Returns:
      a × b

    • one

      public Integer one()
      Description copied from interface: Semiring
      Returns the multiplicative identity (one element).

      Satisfies: 1 × a = a × 1 = a for all elements a.

      Specified by:
      one in interface Semiring<Integer>
      Returns:
      the multiplicative identity

    • isMultiplicationCommutative

      public boolean isMultiplicationCommutative()
      Description copied from interface: Semiring
      Tests whether multiplication is commutative in this semiring.
      Specified by:
      isMultiplicationCommutative in interface Semiring<Integer>
      Returns:
      true if multiplication commutes

    • inverse

      public Integer inverse(Integer element)
      Description copied from interface: Group
      Returns the inverse of the given element.

      For element a, returns a⁻¹ such that: a ∗ a⁻¹ = a⁻¹ ∗ a = e (identity).

      Specified by:
      inverse in interface Group<Integer>
      Parameters:
      element - the element to invert
      Returns:
      the inverse element

    • isCountable

      public boolean isCountable()
      Description copied from interface: InfiniteSet
      Returns true if this set is countable (i.e., its elements can be put in one-to-one correspondence with the natural numbers).

      Examples:

      • Countable: â„• (Natural numbers), ℤ (Integers), â„š (Rational numbers)
      • Uncountable: ℝ (Real numbers), â„‚ (Complex numbers)

      Specified by:
      isCountable in interface InfiniteSet<Integer>
      Returns:
      true if this set is countable

    • contains

      public boolean contains(Integer element)
      Description copied from interface: Set
      Tests whether this set contains the specified element.

      This is the fundamental operation of a set - membership testing.

      Specified by:
      contains in interface Set<Integer>
      Parameters:
      element - the element to test for membership
      Returns:
      true if this set contains the element, false otherwise
      See Also:

    • isEmpty

      public boolean isEmpty()
      Description copied from interface: Set
      Returns true if this set contains no elements.

      The empty set (∅) is a fundamental concept in set theory. It is the unique set containing no elements.

      Specified by:
      isEmpty in interface Set<Integer>
      Returns:
      true if this set is empty

    • description

      public String description()
      Description copied from interface: Set
      Returns a human-readable description of this set.

      Examples:

      • "ℝ (Real Numbers)"
      • "ℤ/12ℤ (Integers modulo 12)"
      • "{1, 2, 3, 4, 5}"

      Specified by:
      description in interface Set<Integer>
      Returns:
      a description of this set

    • toString

      public String toString()
      Overrides:
      toString in class Object