**Monthly Downloads**: 53

**Programming language**: Haskell

**License**: Apache License 2.0

**Latest version**: v0.5.0.0

## README

## partial-semigroup

Packages on Hackage:

## Semigroups (background)

A *semigroup* is a set with a binary associative operator. In Haskell we
represent semigroups as instances of the `Semigroup`

typeclass, which looks
something like this:

```
class Semigroup a where (<>) :: a -> a -> a
```

This was once provided by the
semigroups package, but is
now in the Haskell standard library as of `base 4.9.0.0`

in 2016.

### The semigroup associativity axiom

The semigroup *associativity* axiom is stated as:

```
(a <> b) <> c = a <> (b <> c)
```

## Partial semigroups

A *partial semigroup* can be defined in two equivalent ways:

- As a semigroup where
`<>`

is a*partial function*(that is, we admit the possibility that`x <> y = ⊥`

for some`x`

and`y`

) - As a new kind of algebraic structure where the operation is
*total*(not partial) but returns`Maybe a`

instead of`a`

.

The second definition is the approach we take here (though we will refer back to
this first definition when we discuss the associativity axiom). The
`partial-semigroup`

package defines the `PartialSemigroup`

class, which looks
like this:

```
class PartialSemigroup a where (<>?) :: a -> a -> Maybe a
```

### The partial semigroup associativity axiom

The partial semigroup associativity axiom is a natural adaptation of the
semigroup associativity axiom, with a slight modification to accommodate
the situations wherein `x <> y = ⊥`

. First we'll express the axiom in terms
of `Semigroup`

and `⊥`

, and then we'll rephrase it in terms of
`PartialSemigroup`

.

#### Definition 1: In terms of `Semigroup`

and `⊥`

For all `x`

, `y`

, `z`

:

If

`x <> y ≠ ⊥`

and`y <> z ≠ ⊥`

, then`x <> (y <> z) = ⊥`

if and only if`(x <> y) <> z = ⊥`

, and- where none of the terms are ⊥, the axiom for total semigroups
`x <> (y <> z) = (x <> y) <> z`

must hold.

#### Definition 2: In terms of `PartialSemigroup`

For all `x`

, `y`

, `z`

:

If

`x <>? y = Just xy`

and`y <>? z = Just yz`

, then`x <>? yz = xy <>? z`

.

## Deriving using GHC generics

If a type derives `Generic`

and all of its fields have `PartialSemigroup`

instances, you can get a `PartialSemigroup`

for free.

```
{-# LANGUAGE DeriveGeneric #-}
import Data.PartialSemigroup.Generics
data T
= A String (Either String String)
| B String
deriving (Eq, Generic, Show)
instance PartialSemigroup T where
(<>?) = genericPartialSemigroupOp
```

This gives us an implementation of `<>?`

which combines values only if they have
the same structure.

```
λ> A "s" (Left "x") <>? A "t" (Left "y")
Just (A "st" (Left "xy"))
>>> B "x" <>? B "y"
Just (B "xy")
```

For values that do *not* have the same structure, `<>?`

produces `Nothing`

.

```
>>> A "s" (Left "x") <>? A "t" (Right "y")
Nothing
>>> A "x" (Left "y") <>? B "z"
Nothing
```

## Property testing

The `partial-semigroup-hedgehog`

package provides a
hedgehog property that you can
use to verify the associativity axiom for `PartialSemigroup`

instances.

The `partial-semigroup-test`

package aggregates all of the test packages
(although at the moment there is only one, so this package is rather
superfluous).