Examples of Semigroups
- Empty semigroup: the empty set forms a semigroup with the empty function as the binary operation.
- Semigroup with one element: there is essentially just one, the singleton {a} with operation a · a = a.
- Semigroup with two elements: there are five which are essentially different.
- The set of positive integers with addition.
- Square nonnegative matrices with matrix multiplication.
- Any ideal of a ring with the multiplication of the ring.
- The set of all finite strings over a fixed alphabet Σ with concatenation of strings as the semigroup operation — the so-called "free semigroup over Σ". With the empty string included, this semigroup becomes the free monoid over Σ.
- A probability distribution F together with all convolution powers of F, with convolution as operation. This is called a convolution semigroup.
- A monoid is a semigroup with an identity element.
- A group is a monoid in which every element has an inverse element.
- Transformation semigroups and monoids
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