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1.Semigroups in General

27.04.2020

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@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.
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