What is a semigroup?
Semigroups provides the structure for sequential process, irreversible in time, and computational transitional states. Semigroups must have a binary operation that is associative. \[(a \cdot b)\cdot c = a \cdot (b\cdot c)\]. Semigroups are mostly used to model systems that are not reversible, unlike groups. Sometimes we want a semigroup to be "upgraded" to a monoid that requires an identity element. Sometimes, we want a set of states \[S\] to have a relation \[ R: S \times S \to S \], that is, having two elements \[s,s_1 \in S\] we want to have another element in S. We can denote the operations by parenthesis \[((s,s1),s2)\]. In short, semigroups satisfy the closure and associativity in its relation(operation).The natural numbers are semigroups with both multiplication and addition.
The integer numbers form a semigroup with both multiplication and addition.
Strings form a semigroup with composition or catenation.
Here you'll find these links: Semigroups Groups
Strings
A string is a sequence of symbols of an alphabet. abbbba is a string of the alphabet {a,b}, and 0010010010010010010 is a binary string.
Programming Mahines
Operation Mapping