# Sylvester.AbstractAlgebra 0.2.5

```dotnet add package Sylvester.AbstractAlgebra --version 0.2.5

```
```NuGet\Install-Package Sylvester.AbstractAlgebra -Version 0.2.5

```
This command is intended to be used within the Package Manager Console in Visual Studio, as it uses the NuGet module's version of Install-Package.
```<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

```
```#r "nuget: Sylvester.AbstractAlgebra, 0.2.5"

```
#r directive can be used in F# Interactive and Polyglot Notebooks. Copy this into the interactive tool or source code of the script to reference the package.
```// Install Sylvester.AbstractAlgebra as a Cake Addin

// Install Sylvester.AbstractAlgebra as a Cake Tool
#tool nuget:?package=Sylvester.AbstractAlgebra&version=0.2.5

```

## 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
Paket.Package["Sylvester.AbstractAlgebra"]

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))
``````
``````S "               Nancy"
``````
``````// Is pad a homomorphism?
``````
``````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]]
``````
``````
``````
Product Compatible and additional computed target framework versions.
.NET net5.0 was computed.  net5.0-windows was computed.  net6.0 was computed.  net6.0-android was computed.  net6.0-ios was computed.  net6.0-maccatalyst was computed.  net6.0-macos was computed.  net6.0-tvos was computed.  net6.0-windows was computed.  net7.0 was computed.  net7.0-android was computed.  net7.0-ios was computed.  net7.0-maccatalyst was computed.  net7.0-macos was computed.  net7.0-tvos was computed.  net7.0-windows was computed.  net8.0 was computed.  net8.0-android was computed.  net8.0-browser was computed.  net8.0-ios was computed.  net8.0-maccatalyst was computed.  net8.0-macos was computed.  net8.0-tvos was computed.  net8.0-windows was computed.
.NET Core netcoreapp2.0 was computed.  netcoreapp2.1 was computed.  netcoreapp2.2 was computed.  netcoreapp3.0 was computed.  netcoreapp3.1 was computed.
.NET Standard netstandard2.0 is compatible.  netstandard2.1 was computed.
.NET Framework net45 is compatible.  net451 was computed.  net452 was computed.  net46 was computed.  net461 was computed.  net462 was computed.  net463 was computed.  net47 was computed.  net471 was computed.  net472 was computed.  net48 was computed.  net481 was computed.
MonoAndroid monoandroid was computed.
MonoMac monomac was computed.
MonoTouch monotouch was computed.
Tizen tizen40 was computed.  tizen60 was computed.
Xamarin.iOS xamarinios was computed.
Xamarin.Mac xamarinmac was computed.
Xamarin.TVOS xamarintvos was computed.
Xamarin.WatchOS xamarinwatchos was computed.
Compatible target framework(s)
Included target framework(s) (in package)

### NuGet packages

This package is not used by any NuGet packages.

### GitHub repositories

This package is not used by any popular GitHub repositories.