nLab invertible semigroup


This entry is about semigroups with two-sided inverses. For semigroups with a unary operator ii such that si(s)s=ss \cdot i(s) \cdot s = s and i(s)si(s)=i(s)i(s) \cdot s \cdot i(s) = i(s), see instead at inverse semigroup.



Representation theory



A semigroup that is also an invertible magma.


With only multiplication

An invertible semigroup is a semigroup (G,()():G×GG)(G,(-)\cdot(-):G\times G\to G) such that for every element aAa \in A, left multiplication and right multiplication by aa are both bijections.

With multiplication and inverses

An invertible semigroup is a semigroup (G,()():G×GG)(G,(-)\cdot(-):G\times G\to G) with a unary operation called the inverse () 1:GG(-)^{-1}:G \to G such that

  • ab 1b=aa \cdot b^{-1} \cdot b = a
  • b 1ba=ab^{-1} \cdot b \cdot a = a
  • bb 1a=ab \cdot b^{-1} \cdot a = a
  • abb 1=aa \cdot b \cdot b^{-1} = a

for all a,bGa,b \in G.

Torsor-like definition

There is an alternate definition of an invertible semigroup that looks like the usual definition of a torsor or heap:

An invertible semigroup is a set SS with a binary operation ()():S×SS(-)\cdot(-):S\times S\to S called multiplication and a unary operation () 1:SS(-)^{-1}:S\to S called inverse satisfying the following laws:

  • associativity: a(bc)=(ab)ca \cdot (b \cdot c) = (a \cdot b) \cdot c for all a,b,cSa,b,c\in S
  • left Malcev identity: bb 1a=ab \cdot b^{-1} \cdot a = a for all a,bSa,b\in S
  • right Malcev identity: ab 1b=aa \cdot b^{-1} \cdot b = a for all a,bSa,b\in S
  • commutativity with inverse elements: aa 1=a 1aa \cdot a^{-1} = a^{-1} \cdot a for all aSa\in S


Every invertible semigroup GG has a pseudo-torsor, or associative Malcev algebras, t:G 3Gt:G^3\to G defined as t(x,y,z)=xy 1zt(x,y,z)=x\cdot y^{-1}\cdot z. If the invertible semigroup is inhabited, then those pseudo-torsors are actually torsors or heaps.


  • Every invertible semigroup is either a group or the empty semigroup.

Last revised on May 23, 2023 at 05:21:40. See the history of this page for a list of all contributions to it.