# 1.Semigroups in General

## 27.04.2020

### 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:

1. 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.