nLab semigroup

Idea

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.

Definition

A semigroup is, equivalently,

• a set equipped with an associative binary operation.

• an associative magma;

• the hom-set of a semicategory with a single object.

Properties

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.

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 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.

Exercise

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.

Examples

• 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 $n:\mathbb{N}$.

Internalization

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

Related topics

• 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?.

• inverse semigroups

• associative quasigroup

• cancellative semigroup?

• weakly reductive semigroup

• one-parameter semigroup

• n-ary semigroup

• semiring

• semi-category, semi-simplicial set, semi-Segal space

• rectangular band

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

References

On the history of the notion:

• Christopher Hollings, The Early Development of the Algebraic Theory of Semigroups, Archive for History of Exact Sciences 63 (2009) 497–536 [doi:10.1007/s00407-009-0044-3]

Semicategories and semigroups are mentioned for instance

• W. Dale Garraway, Section 2 in Sheaves for an involutive quantaloid, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 4 (2005), p. 243-274 (numdam)

Discussion in the context of Lie theory:

• Joachim Hilgert, Karl-Hermann Neeb, Lie Semigroups and their Applications. Lecture Notes in Mathematics 1552, Springer 1993 (doi:10.1007/BFb0084640)

category: algebra