Sylvester.AbstractAlgebra 0.2.5

F# Library for defining, exploring and proving concepts in abstract algebra.

Install-Package Sylvester.AbstractAlgebra -Version 0.2.5
dotnet add package Sylvester.AbstractAlgebra --version 0.2.5
<PackageReference Include="Sylvester.AbstractAlgebra" Version="0.2.5" />
For projects that support PackageReference, copy this XML node into the project file to reference the package.
paket add Sylvester.AbstractAlgebra --version 0.2.5
The NuGet Team does not provide support for this client. Please contact its maintainers for support.

Sylvester.AbstractAlgebra

The Sylvester abstract algebra library contains types and operations for rigorously defining abstract algebra structures and concepts.

// Use the Sylvester abstract algebra package
#load "Paket.fsx"
Paket.Package["Sylvester.AbstractAlgebra"] 
#load "Paket.Generated.Refs.fsx"

open System 
open System.Linq
open Sylvester

Morphisms

// Define a custom symbol type S with a (+) operator and zero
// We could just also use plain strings
type S = S of string with
    static member (+) (S l, S r) = S (l + r)
    static member Zero = S ""

// Define an infinite sequence of L strings
let Sym = infiniteSeq ((+) 65 >> Char.ConvertFromUtf32 >> S)
Sym
// Define a monoid using our set and + operator and zero element
let L = Monoid(Sym, (+), S.Zero)
L
seq
  [(S "A", S "B", S "AB"); (S "B", S "C", S "BC"); (S "C", S "D", S "CD");
   (S "D", S "E", S "DE"); ...]
// Create 2 S values
let a, b = S "Nancy", S "Drew"
a + b
S "NancyDrew"
// Create a L morphism using the PadLeft string function
let Pad = Morph(L, fun l -> let (S s) = l in S(s.PadLeft 20))
let pad = Pad.Map
pad a
S "               Nancy"
// Is pad a homomorphism?
pad a + pad b
S "               Nancy                Drew"
pad (a + b)
S "           NancyDrew"
pad a + pad b = pad (a + b)
false

Rings

Zpos
seq [(0, 1, 1); (1, 2, 3); (2, 3, 5); (3, 4, 7); ...]

Subsets

let s = seq{1..6} |> Set.ofSubsets
s
Seq
  [|Empty; Seq [|1|]; Seq [|2|]; Seq [|1; 2|]; Seq [|3|]; Seq [|1; 3|];
    Seq [|2; 3|]; Seq [|1; 2; 3|]; Seq [|4|]; Seq [|1; 4|]; Seq [|2; 4|];
    Seq [|1; 2; 4|]; Seq [|3; 4|]; Seq [|1; 3; 4|]; Seq [|2; 3; 4|];
    Seq [|1; 2; 3; 4|]; Seq [|5|]; Seq [|1; 5|]; Seq [|2; 5|]; Seq [|1; 2; 5|];
    Seq [|3; 5|]; Seq [|1; 3; 5|]; Seq [|2; 3; 5|]; Seq [|1; 2; 3; 5|];
    Seq [|4; 5|]; Seq [|1; 4; 5|]; Seq [|2; 4; 5|]; Seq [|1; 2; 4; 5|];
    Seq [|3; 4; 5|]; Seq [|1; 3; 4; 5|]; Seq [|2; 3; 4; 5|];
    Seq [|1; 2; 3; 4; 5|]; Seq [|6|]; Seq [|1; 6|]; Seq [|2; 6|];
    Seq [|1; 2; 6|]; Seq [|3; 6|]; Seq [|1; 3; 6|]; Seq [|2; 3; 6|];
    Seq [|1; 2; 3; 6|]; Seq [|4; 6|]; Seq [|1; 4; 6|]; Seq [|2; 4; 6|];
    Seq [|1; 2; 4; 6|]; Seq [|3; 4; 6|]; Seq [|1; 3; 4; 6|]; Seq [|2; 3; 4; 6|];
    Seq [|1; 2; 3; 4; 6|]; Seq [|5; 6|]; Seq [|1; 5; 6|]; Seq [|2; 5; 6|];
    Seq [|1; 2; 5; 6|]; Seq [|3; 5; 6|]; Seq [|1; 3; 5; 6|]; Seq [|2; 3; 5; 6|];
    Seq [|1; 2; 3; 5; 6|]; Seq [|4; 5; 6|]; Seq [|1; 4; 5; 6|];
    Seq [|2; 4; 5; 6|]; Seq [|1; 2; 4; 5; 6|]; Seq [|3; 4; 5; 6|];
    Seq [|1; 3; 4; 5; 6|]; Seq [|2; 3; 4; 5; 6|]; Seq [|1; 2; 3; 4; 5; 6|]|]

Lattices


let lat = Lattice(s, (|+|), (|*|))
lat
Sylvester.Lattice`1[Sylvester.Set`1[System.Int32]]

Sylvester.AbstractAlgebra

The Sylvester abstract algebra library contains types and operations for rigorously defining abstract algebra structures and concepts.

// Use the Sylvester abstract algebra package
#load "Paket.fsx"
Paket.Package["Sylvester.AbstractAlgebra"] 
#load "Paket.Generated.Refs.fsx"

open System 
open System.Linq
open Sylvester

Morphisms

// Define a custom symbol type S with a (+) operator and zero
// We could just also use plain strings
type S = S of string with
    static member (+) (S l, S r) = S (l + r)
    static member Zero = S ""

// Define an infinite sequence of L strings
let Sym = infiniteSeq ((+) 65 >> Char.ConvertFromUtf32 >> S)
Sym
// Define a monoid using our set and + operator and zero element
let L = Monoid(Sym, (+), S.Zero)
L
seq
  [(S "A", S "B", S "AB"); (S "B", S "C", S "BC"); (S "C", S "D", S "CD");
   (S "D", S "E", S "DE"); ...]
// Create 2 S values
let a, b = S "Nancy", S "Drew"
a + b
S "NancyDrew"
// Create a L morphism using the PadLeft string function
let Pad = Morph(L, fun l -> let (S s) = l in S(s.PadLeft 20))
let pad = Pad.Map
pad a
S "               Nancy"
// Is pad a homomorphism?
pad a + pad b
S "               Nancy                Drew"
pad (a + b)
S "           NancyDrew"
pad a + pad b = pad (a + b)
false

Rings

Zpos
seq [(0, 1, 1); (1, 2, 3); (2, 3, 5); (3, 4, 7); ...]

Subsets

let s = seq{1..6} |> Set.ofSubsets
s
Seq
  [|Empty; Seq [|1|]; Seq [|2|]; Seq [|1; 2|]; Seq [|3|]; Seq [|1; 3|];
    Seq [|2; 3|]; Seq [|1; 2; 3|]; Seq [|4|]; Seq [|1; 4|]; Seq [|2; 4|];
    Seq [|1; 2; 4|]; Seq [|3; 4|]; Seq [|1; 3; 4|]; Seq [|2; 3; 4|];
    Seq [|1; 2; 3; 4|]; Seq [|5|]; Seq [|1; 5|]; Seq [|2; 5|]; Seq [|1; 2; 5|];
    Seq [|3; 5|]; Seq [|1; 3; 5|]; Seq [|2; 3; 5|]; Seq [|1; 2; 3; 5|];
    Seq [|4; 5|]; Seq [|1; 4; 5|]; Seq [|2; 4; 5|]; Seq [|1; 2; 4; 5|];
    Seq [|3; 4; 5|]; Seq [|1; 3; 4; 5|]; Seq [|2; 3; 4; 5|];
    Seq [|1; 2; 3; 4; 5|]; Seq [|6|]; Seq [|1; 6|]; Seq [|2; 6|];
    Seq [|1; 2; 6|]; Seq [|3; 6|]; Seq [|1; 3; 6|]; Seq [|2; 3; 6|];
    Seq [|1; 2; 3; 6|]; Seq [|4; 6|]; Seq [|1; 4; 6|]; Seq [|2; 4; 6|];
    Seq [|1; 2; 4; 6|]; Seq [|3; 4; 6|]; Seq [|1; 3; 4; 6|]; Seq [|2; 3; 4; 6|];
    Seq [|1; 2; 3; 4; 6|]; Seq [|5; 6|]; Seq [|1; 5; 6|]; Seq [|2; 5; 6|];
    Seq [|1; 2; 5; 6|]; Seq [|3; 5; 6|]; Seq [|1; 3; 5; 6|]; Seq [|2; 3; 5; 6|];
    Seq [|1; 2; 3; 5; 6|]; Seq [|4; 5; 6|]; Seq [|1; 4; 5; 6|];
    Seq [|2; 4; 5; 6|]; Seq [|1; 2; 4; 5; 6|]; Seq [|3; 4; 5; 6|];
    Seq [|1; 3; 4; 5; 6|]; Seq [|2; 3; 4; 5; 6|]; Seq [|1; 2; 3; 4; 5; 6|]|]

Lattices


let lat = Lattice(s, (|+|), (|*|))
lat
Sylvester.Lattice`1[Sylvester.Set`1[System.Int32]]

Release Notes

Update to latest Sylph version.

NuGet packages

This package is not used by any NuGet packages.

GitHub repositories

This package is not used by any popular GitHub repositories.

Version History

Version Downloads Last updated
0.2.5 187 5/3/2020
0.2.3.2 144 3/24/2020
0.2.3.1 133 3/24/2020
0.2.3 149 3/24/2020
0.2.1.2 190 1/29/2020
0.2.1.1 181 1/29/2020
0.2.1 181 1/29/2020