nLab
semigroup

Contents

Contents

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,

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 𝒞\mathcal{C} can be thought of as a semigroup by defining S=Mor(𝒞){0}S = \text{Mor}(\mathcal{C})\cup \{0\} and taking f*g=fgf*g = f \circ g for any composable morphisms f,gf, g, and f*g=0f*g = 0 otherwise. Then the semigroup (S,*)(S, *) fully describes 𝒞\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 groupoid?s. Inverse semigroups naturally occur when considering partial symmetries of an object.

Examples

  • The natural numbers with a binary operation ()+ n():×(-) +_n (-):\mathbb{N}\times\mathbb{N}\to\mathbb{N} inductively defined as 0+ n0=n0 +_n 0 = n, S(x)+ ny=S(x+ ny)S(x) +_n y = S(x +_n y), and x+ nS(y)=S(x+ ny)x +_n S(y) = S(x +_n y) for all x,y:x,y:\mathbb{N} is a commutative semigroup for all n:n:\mathbb{N}.

Internalization

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

algebraic structureoidification
truth valuetransitive relation
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
(left,right) cancellative monoid(left,right) cancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
differential ring?differential ringoid?
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
strict monoidal categorystrict 2-category
strict 2-groupstrict 2-groupoid
monoidal poset?2-poset
monoidal groupoid?(2,1)-category
monoidal category2-category/bicategory
2-group2-groupoid/bigroupoid

References

Semicategories and semigroups are mentioned for instance in section 2 in

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

Last revised on May 25, 2021 at 00:16:27. See the history of this page for a list of all contributions to it.