algebra-ring

defines an algebra ring structure

Installation

With npm do

npm install algebra-ring

Example

All code in the examples below is intended to be contained into a single file.

Real

Create a ring structure over real numbers.

const ring = require('algebra-ring')

// Define operators.
function contains (a) {
  // NaN, Infinity and -Infinity are not allowed
  return (typeof a === 'number' && isFinite(a))
}

function equality (a, b) { return a === b }

function addition (a, b) { return a + b }

function negation (a) { return -a }

function multiplication (a, b) { return a * b }

function inversion (a) { return 1 / a }

// Create a ring by defining its identities and operators.
const R = ring([0, 1], {
  equality: equality,
  contains: contains,
  addition: addition,
  negation: negation,
  multiplication: multiplication,
  inversion: inversion
})

You get a Ring that is a Group with multiplication operator. The multiplication operator must be closed respect the underlying set; its inverse operator is division.

This is the list of ring operators:

contains
notContains
equality
disequality
addition
subtraction
negation
multiplication
division
inversion

The neutral element for addition and multiplication are, as usual, called zero and one respectively.

R.contains(10) // true
R.contains(-1, 0.5, Math.PI, 5) // true
R.notContains(Infinity) // true

R.addition(1, 2) // 3
R.addition(2, 3, 5, 7) // 17

R.equality(R.negation(2), -2) // true

R.subtraction(2, 3) // -1

R.multiplication(2, 5) // 10
R.multiplication(2, 2, 2, 2) // 16

R.equality(R.inversion(10), 0.1) // true

R.division(1, 2) // 0.5

R.equality(R.addition(2, R.zero), 2) // true
R.equality(R.subtraction(2, 2), R.zero) // true

R.equality(R.multiplication(2, R.one), 2) // true
R.equality(R.division(2, 2), R.one) // true

R.division(1, 0) // will complain
R.inversion(R.zero) // will complain too

Boolean

It is possible to create a ring over the booleans.

const Boole = ring([false, true], {
  equality: (a, b) => (a === b),
  contains: (a) => (typeof a === 'boolean'),
  addition: (a, b) => (a || b),
  negation: (a) => (a),
  multiplication: (a, b) => (a && b),
  inversion: (a) => (a)
})

There are only two elements, you know, true and false.

Boole.contains(true, false) // true

Check that false is the neutral element of addition and true is the neutral element of multiplication.

Boole.addition(true, Boole.zero) // true
Boole.multiplication(true, Boole.one) // true

As usual, it is not allowed to divide by zero: the following code will throw.

Boole.division(true, false)
Boole.inversion(Bool.zero)

API

`ring(identities, operator)`

@param {Array} identities
@param {*} identities[0] a.k.a zero
@param {*} identities1 a.k.a one
@param {Object} operator
@param {Function} operator.contains
@param {Function} operator.equality
@param {Function} operator.addition
@param {Function} operator.negation
@param {Function} operator.multiplication
@param {Function} operator.inversion
@returns {Object} ring

`ring.error`

An object exposing the following error messages:

cannotDivideByZero
doesNotContainIdentity
identityIsNotNeutral

License

MIT