Class Octonions

java.lang.Object

org.episteme.core.mathematics.sets.Octonions

All Implemented Interfaces:

AbelianGroup<Octonion>, AbelianMonoid<Octonion>, Group<Octonion>, Magma<Octonion>, Monoid<Octonion>, Field<Octonion>, Ring<Octonion>, Semiring<Octonion>, InfiniteSet<Octonion>, Set<Octonion>

public final class Octonions
extends Object
implements Field<Octonion>, InfiniteSet<Octonion>

The structure of Octonions (O).

This class represents the structure of octonions. It implements Field as requested, though technically O is a non-associative division algebra.

Note: Octonion multiplication is non-commutative and non-associative.

* @author Silvere Martin-Michiellot

Since:

1.0

Author:

Gemini AI (Google DeepMind)

Field Summary

Fields

Modifier and Type	Field	Description
static final Octonions	INSTANCE	Singleton instance

Method Summary

Modifier and Type	Method	Description
Octonion	add(Octonion a, Octonion b)	Returns the sum of two elements (additive notation).
int	characteristic()	Returns the characteristic of this field.
boolean	contains(Octonion element)	Tests whether this set contains the specified element.
String	description()	Returns a human-readable description of this set.
static Octonions	getInstance()	Returns the singleton instance.
Octonion	inverse(Octonion element)	Returns the multiplicative inverse of a non-zero element.
boolean	isCountable()	Returns true if this set is countable (i.e., its elements can be put in one-to-one correspondence with the natural numbers).
boolean	isEmpty()	Returns true if this set contains no elements.
boolean	isMultiplicationCommutative()	Fields always have commutative multiplication.
Octonion	multiply(Octonion a, Octonion b)	Returns the product of two elements.
Octonion	negate(Octonion element)	Returns the additive inverse (negation) of an element.
Octonion	one()	Returns the multiplicative identity (one element).
Octonion	operate(Octonion a, Octonion b)	Performs the binary operation on two elements.
String	toString()
Octonion	zero()	Returns the additive identity (zero element).

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Methods inherited from interface AbelianGroup

isCommutative, subtract

Methods inherited from interface AbelianMonoid

identity, isCommutative

Methods inherited from interface Field

divide, hasUnity

Methods inherited from interface Monoid

isAssociative

Methods inherited from interface Semiring

pow

Field Details

INSTANCE

public static final Octonions INSTANCE

Singleton instance

Method Details

getInstance

public static Octonions getInstance()

Returns the singleton instance.

Returns:

the Octonions structure

operate

public Octonion operate(Octonion a,
 Octonion 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<Octonion>

Parameters:

a - the first operand

b - the second operand

Returns:

the result of a ∗ b

See Also:

• Magma.isAssociative()
• Magma.isCommutative()

add

public Octonion add(Octonion a,
 Octonion 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<Octonion>

Parameters:

a - the first addend

b - the second addend

Returns:

a + b

characteristic

public int characteristic()

Description copied from interface: Field

Returns the characteristic of this field.

The characteristic is the smallest positive integer n such that:
1 + 1 + ... + 1 (n times) = 0
If no such n exists, the characteristic is 0.

Examples:

• char(â„š) = char(ℝ) = char(â„‚) = 0
• char(𝔽ₚ) = p (for prime p)

Specified by:

characteristic in interface Field<Octonion>

Returns:

the characteristic (0 for infinite fields, p for finite fields)

zero

public Octonion zero()

Description copied from interface: AbelianMonoid

Returns the additive identity (zero element).

Delegates to AbelianMonoid.identity().

Specified by:

zero in interface AbelianMonoid<Octonion>

Returns:

the zero element

negate

public Octonion negate(Octonion 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<Octonion>

Parameters:

element - the element to negate

Returns:

-element

See Also:

• Group.inverse(Object)

multiply

public Octonion multiply(Octonion a,
 Octonion 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<Octonion>

Parameters:

a - the first factor

b - the second factor

Returns:

a × b

one

public Octonion 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<Octonion>

Returns:

the multiplicative identity

isMultiplicationCommutative

public boolean isMultiplicationCommutative()

Description copied from interface: Field

Fields always have commutative multiplication.

Specified by:

isMultiplicationCommutative in interface Field<Octonion>

Specified by:

isMultiplicationCommutative in interface Semiring<Octonion>

Returns:

always true

inverse

public Octonion inverse(Octonion element)

Description copied from interface: Field

Returns the multiplicative inverse of a non-zero element.

For element a ≠0, returns a⁻¹ such that: a × a⁻¹ = a⁻¹ × a = 1

Examples:

• In â„š: inverse(2/3) = 3/2
• In ℝ: inverse(5.0) = 0.2
• In â„‚: inverse(3+4i) = (3-4i)/25

Specified by:

inverse in interface Field<Octonion>

Specified by:

inverse in interface Group<Octonion>

Parameters:

element - the element to invert (must be non-zero)

Returns:

the multiplicative inverse

See Also:

• Field.divide(Object, Object)
• Semiring.one()

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<Octonion>

Returns:

true if this set is countable

contains

public boolean contains(Octonion 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<Octonion>

Parameters:

element - the element to test for membership

Returns:

true if this set contains the element, false otherwise

See Also:

• Set.isEmpty()
• invalid reference

  #size()

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<Octonion>

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<Octonion>

Returns:

a description of this set

toString

public String toString()

Overrides:

toString in class Object