Semigroup - Examples of Semigroups

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