nLab
semigroup

A semigroup is like a monoid where there might not be an identity element. In other words, a semigroup is a set equipped with an associative binary operation. Equivalently, we may define a semigroup to be an associative magma.

Some semigroups happen to be monoids; even then, a semigroup homomorphism might not be a monoid homomorphism (because it might not preserve the identity element). Nevertheless, semigroup isomorphisms must be monoid isomorphisms. Thus, the identity element of a monoid forms a property-like structure on the underlying semigroup.

As a monoid is a category with one object, so a semigroup is a semicategory? with one object.

Attitudes Toward Semigroups

Some mathematicians consider semigroups to be a case of centipede mathematics. Category theorists sometimes look with scorn on semigroups, because unlike a monoid, a semigroup is not an example of a category.

However, a semigroup can be promoted to a monoid by adjoining a new element and decreeing it to be the identity. This gives a full and faithful functor from the category of semigroups to the category of monoids. So, a semigroup can actually be seen as a monoid with an extra property.

Exercise

What is this property?

On the other hand, analysts run across semigroups often in the wild, and don't always want to add formal identities just to turn them into moniods.

Another variant with strong links with category theory is that of inverse semigroup?s, which Charles Ehresmann showed were closely related to ordered groupoid?s. Inverse semigroups naturally occur when considering partial symmetries of an object.

(For the moment, look at:

• Lawson, Mark V. Constructing ordered groupoids. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 2 (2005), p. 123–138, URL stable

or his book:

• M.V. Lawson, Inverse semigroups: the theory of partial symmetries, World Scientific, 1998.

This will need expanding sometime.)

Internalization

We can internalize the concept of semigroup in any monoidal category (or even multicategory) $V$ to get a semigroup object in $V$.