A semigroup is an algebraic structure $(S, \otimes)$ in which $S$ is a non-empty set and $\otimes : S \times S \rightarrow S$ is a binary associative operation on $S$, such that the equation $(a \otimes b) \otimes c = a \otimes (b \otimes c)$ holds for all $a, b, c \in \mathcal{S}$. In category theory, a semigroup is a monoid where there might not be an identity element. More formally, a semigroup is a semicategory (a category without the requirement for identiy morphism) $\mathcal{C}$ with just one object $S$ and the following conditions:

• The set of morphisms (hom-set) is closed under composition: For every pair of morphisms $f, g$ in $Hom_\mathcal{C}(S,S)$, their composition $f \circ g$ also belongs to $Hom_{C}(S,S)$

• The composition operation is associative: For any three morphisms $f, g, h$ in $Hom_{C}(S, S)$, we have $(f \circ g) \circ h = f \circ (g \circ h)$.

Example

A type qualifies as a Semigroup if it offers an associative function (<>), allowing the merging of any two type values into a single one.

Haskell Definition of Semigroup (Interface)

class Semigroup a where
    -- ⊗  :  S x S  -> S (multiplication)
    (<>) :: a -> a -> a

Associativity implies that the following condition must always hold:

(a <> b) <> c == a <> (b <> c)

An Instance of Semigroup, the List Semigroup

instance Semigroup [a] where
    (<>) = (++)

Another Instance, the Maybe Semigroup

instance Semigroup a => Semigroup (Maybe a) where
    Just a  <> Just b  = Just (a <> b)
    Just a  <> Nothing = Just a
    Nothing <> Just b  = Just b
    Nothing <> Nothing = Nothing

All of the above is already implemented in the standard Haskell library, so you can also simply open an interactive Haskell interpreter (ghci) and test the following examples.

ghci> [1,2,3] <> [4,5,6]
[1,2,3,4,5,6]

ghci> Just "A" <> Just "B"
Just "AB"

ghci> Nothing <> Just "A" <> Nothing
Just "A"

ghci> Nothing <> Nothing
Nothing

Some more examples are:

• The naturals numbers without zero $({\mathbb {N}},+)$ under addition. This forms a semigroup because addition is associative.

• The natural numbers $({\mathbb {N}}_{0}, \times )$ under multiplication. This forms a semigroup because multiplication is associative, but is also a monoid $\left({\mathbb {N}}_{0},\times,1\right)$ since 1 serves as the identity, as any number multiplied by 1 remains the same.

• Non-empty strings $(\texttt{String},++)$ under concatenation. This forms a semigroup because string concatenation is associative.

References