Semigroups
What is a Semigroup?
A semigroup is an algebraic structure (S,*) that consists of a set S and a binary operation S×S→S, called the composition operation. This operation satisfies the associative property, meaning that (a∗b)∗c=a∗(b∗c) ∀ a,b,c ∈S
It can also be referred to as a pseudogroup.
Semigroups do not require the presence of a neutral or inverse element.
If the semigroup includes a neutral element, it is known as a monoid.
Example
Take, for instance, the set of natural numbers N combined with the addition operation +.
(N,+)
This set-up forms a groupoid because it involves an internal binary operation.
Moreover, it also constitutes a semigroup because the operation is associative.
For any three natural numbers a, b, and c, addition is associative:
a+(b+c)=(a+b)+c
For example, if a=2, b=3, c=4:
2+(3+4)=(2+3)+4=9
And so forth.