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The term semigroup does not imply any of these assumptions. en.wikipedia.org

In the language of category theory, any semigroup with one element is a terminal object in the category of semigroups. en.wikipedia.org

In spite of its extreme triviality, the semigroup with one element is important in many situations. en.wikipedia.org

Here a semigroup need not have an identity element; a monoid is a semigroup with an identity element (also called unit). en.wikipedia.org

Conversely, the kernel of any semigroup homomorphism is a semigroup congruence. en.wikipedia.org

This is enough to form the real numbers into a commutative semigroup. en.wikipedia.org

A semigroup is said to be periodic if all of its elements are of finite order. en.wikipedia.org

This ensures that principal ideals generated by some semigroup element do indeed contain that element. en.wikipedia.org

Applications outside of the semigroup and monoid theories are now computationally feasible. en.wikipedia.org

Outside semigroup theory, a unique inverse as defined in this section is sometimes called a quasi-inverse. en.wikipedia.org

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