algebra-group

defines an algebra group structure

NPM version Build Status Dependency Status JavaScript Style Guide

Table Of Contents

Installation

With npm do

npm install algebra-group

Examples

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

Integer additive group

Create the Integer additive group.

const algebraGroup = require('algebra-group')

// Define identity element.
const zero = 0

// Define operators.
function isInteger (a) {
  // NaN, Infinity and -Infinity are not allowed
  return (typeof n === 'number') && isFinite(n) && (n % 1 === 0)
}

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

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

function negation (a) { return -a }

// Create Integer additive group a.k.a (Z, +).
const Z = algebraGroup({
  identity: zero,
  contains: isInteger,
  equality: equality,
  compositionLaw: addition,
  inversion: negation
})

You get a group object with zero as identity and the following group operators:

Z.contains(2) // true
Z.contains(2.5) // false
Z.contains('xxx') // false
Z.notContains(false) // true
Z.notContains(Math.PI) // true
Z.contains(-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) // true
Z.contains(1, 2, 3, 4.5) // false, 4.5 is not an integer

Z.addition(1, 2) // 3
Z.addition(1, 2, 3, 4) // 10

Z.negation(5) // -5

Z.subtraction(5, 1) // 4
Z.subtraction(5, 1, 1, 1, 1, 1) // 0

Z.equality(Z.subtraction(2, 2), Z.zero) // true

R\{0} multiplicative group

Consider R\{0}, the set of Real numbers minus 0, with multiplication as composition law.

It is necessary to remove 0, otherwise there is an element which inverse does not belong to the group, which breaks group laws.

It makes sense to customize group props, which defaults to additive group naming.

function isRealAndNotZero (n) {
  // NaN, Infinity and -Infinity are not allowed
  return (typeof n === 'number') && (n !== 0) && isFinite(n)
}

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

function inversion (a) { return 1 / a }

// Create Real multiplicative group a.k.a (R, *).

const R = algebraGroup({
  identity: 1,
  contains: isRealAndNotZero,
  equality: equality,
  compositionLaw : multiplication,
  inversion: inversion
}, {
  compositionLaw: 'multiplication',
  identity: 'one',
  inverseCompositionLaw: 'division',
  inversion: 'inversion'
})

You get a group object with one as identity and the following group operators:

R.contains(10) // true
R.contains(Math.PI, Math.E, 1.7, -100) // true
R.notContains(Infinity) // true

R.inversion(2) // 0.5

// 2 * 3 * 5 = 30 = 60 / 2
R.equality(R.multiplication(2, 3, 5), R.division(60, 2)) // true

R+ multiplicative group

Create the multiplicative group of positive real numbers (0,∞).

It is a well defined group, since

Let’s customize group props, with a shorter naming.

function isRealAndPositive (n) {
  // NaN, Infinity are not allowed
  return (typeof n === 'number') && (n > 0) && isFinite(n)
}

const Rp = algebraGroup({
  identity: 1,
  contains: isRealAndPositive,
  equality: equality,
  compositionLaw: multiplication,
  inversion: inversion
}, {
  compositionLaw: 'mul',
  equality: 'eq',
  disequality: 'ne',
  identity: 'one',
  inverseCompositionLaw: 'div',
  inversion: 'inv'
})

You get a group object with one identity and the following group operators:

Rp.contains(Math.PI) // true
Rp.notContains(-1) // true
Rp.eq(Rp.inv(4), Rp.div(Rp.one, 4)) // true
Rp.mul(2, 4) // 8

API

group(identity, operator)

group.error

An object exposing the following error messages:

For example, the following snippets will throw.

argumentIsNotInGroup

R.inversion(0) // 0 is not in group R\{0}
Rp.mul(1, 0.1, -1, 0.5) // -1 is not in R+

equalityIsNotReflexive

algebraGroup({
  identity: 1,
  contains: isRealAndNotZero,
  equality: function (a, b) { return a > b }, // not well defined
  compositionLaw: multiplication,
  inversion: inversion
})

identityIsNotInGroup

algebraGroup({
  identity: -1,
  contains: isRealAndPositive,
  equality: equality,
  compositionLaw: multiplication,
  inversion: inversion
})

identityIsNotNeutral

algebraGroup({
  identity: 2,
  contains: isRealAndNotZero,
  equality: equality,
  compositionLaw: multiplication,
  inversion: inversion
})

License

MIT

Contributors