Contents/Index
@1. 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.