Semigroups in General

Definition

The general theory on semigroups. A semigroup is a tuple \( S = (M,\star) \) of a set \( M \) and a binary operator \( \star \) for which the following axiom holds:

Associativity: For every \( a,b,c \in S \) we have that \( a \star (b \star c) = (a \star b) \star c \).

Since no identity element is required, a semigroup can be empty.

Examples

As stated we can form a semigroup of the empty set and the empty function as operator \( S = (\emptyset,ety) \).
The set \( \mathbb{N} = \{0, 1, 2, ...\} \) with the operator \( \cdot \) forms a semigroup. Call it \( S = (\mathbb{N},\cdot) \). This is not a monoid, since for no \( a \in S \) we have that \( a \cdot 0 = 0 \cdot a = 1 \).
Observe the set \( \mathbb{N}^{+} = \{1,2,...\} \). This forms the semigroup \( S = (\mathbb{N}^{+},0) \). If we add \( 0 \) to the set, \( S \) becomes a monoid.