+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Algebra +-- {: .hide} [[!include algebra - contents]] =-- =-- =-- # Contents * table of contents {:toc} ## 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 [[associativity|associative]] [[binary operation]]. * an [[associative magma]]; * the [[hom-set]] of a [[semicategory]] with a single [[object]]. ## Properties Some semigroups happen to be [[monoid]]s; 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 [[group]]s 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 theory|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 [[faithful functor]] from the category of semigroups to the category of monoids. So, a semigroup can actually be seen as a monoid with extra structure. +-- {: .un_example} ###### Exercise Describe this structure. =-- 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 * A left or right [[ideal in a monoid|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 [[action]]s on sets - its right [[M-Set]] $= Set^{M^op}$. The set of all of $M$'s right ideals corresponds to the elements of the [[subobject classifier|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 [[internalization|internalize]] the concept of semigroup in any [[monoidal category]] (or even [[multicategory]]) $V$ to get a __semigroup object__ in $V$. ## Related concepts * [[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]] * [[band]], [[rectangular band]] [[!include oidification - table]] ## References On the history of the notion: * {#Hollings09} 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](https://doi.org/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](http://www.numdam.org/item?id=CTGDC_2005__46_4_243_0)) 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](https://link.springer.com/book/10.1007/BFb0084640)) category: algebra [[!redirects semigroup]] [[!redirects semigroups]] [[!redirects semigroup object]] [[!redirects semigroup objects]] [[!redirects semi-group]] [[!redirects semi-groups]] [[!redirects semi-monoid]] [[!redirects semi-monoids]] [[!redirects semimonoid]] [[!redirects semimonoids]] [[!redirects Lie semigroup]] [[!redirects Lie semigroups]]