A semigroup is like a monoid where there might not be an identity element.
The term “semigroup” is standard, but semi-monoid would be more systematic.
A semigroup is, equivalently,
a set equipped with an associative binary operation.
the hom-set of a semicategory with a single object.
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.
This should be contrasted with the phenomenon that a semigroup homomorphism between two semigroups that happens to be groups does, in fact, happen to be a group homomorphism, since in this special case one can show a semigroup homomorphism must preserve the identity and inverses.
As a monoid is a category with one object, so a semigroup is a semicategory with one object.
Any small category $\mathcal{C}$ can be thought of as a semigroup by defining $S = \text{Mor}(\mathcal{C})\cup \{0\}$ and taking $f*g = f \circ g$ for any composable morphisms $f, g$, and $f*g = 0$ otherwise. Then the semigroup $(S, *)$ fully describes $\mathcal{C}$. This type of semigroup is a weakly reductive semigroup.
Generalizing this, any category can be thought of as a semigroup which isn’t necessarily defined on a set.
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 fully faithful functor from the category of semigroups to the category of monoids. So, a semigroup can actually be seen as a monoid with extra property.
Describe 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 monoids.
Another variant with strong links with category theory is that of inverse semigroups, which Charles Ehresmann showed were closely related to ordered groupoids. Inverse semigroups naturally occur when considering partial symmetries of an object.
A left or right ideal of a monoid $M$ is a subsemigroup of $M$ and is only a submonoid if it contains the unit in which case it is $M$ itself. A monoid $M$ induces the topos of its right actions on sets - its right M-Set $= Set^{M^op}$. The set of all of $M$‘s right ideals corresponds to the elements of the truth value object, $\Omega$, of this topos. The analogous construction holds for left M-Sets $= Set^{M}$ .
The natural numbers with a binary operation $(-) +_n (-):\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ inductively defined as $0 +_n 0 = n$, $S(x) +_n y = S(x +_n y)$, and $x +_n S(y) = S(x +_n y)$ for all $x,y:\mathbb{N}$ is a commutative semigroup for all $n:\mathbb{N}$.
We can internalize the concept of semigroup in any monoidal category (or even multicategory) $V$ to get a semigroup object in $V$.
magma (nonassociative version)
commutative semigroup (commutative version)
invertible semigroup (invertible version)
monoid (unital version)
There is a strong connection between semigroups and finite automata; see e.g. Krohn-Rhodes theory?.
cancellative semigroup?
On the history of the notion:
Semicategories and semigroups are mentioned for instance
Discussion in the context of Lie theory:
Last revised on August 23, 2024 at 15:38:30. See the history of this page for a list of all contributions to it.