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

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

With multiplication and inverses

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

$a \cdot b^{-1} \cdot b = a$

$b^{-1} \cdot b \cdot a = a$

$b \cdot b^{-1} \cdot a = a$

$a \cdot b \cdot b^{-1} = a$

for all $a,b \in G$.

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 $S$ with a binary operation $(-)\cdot(-):S\times S\to S$ called multiplication and a unary operation $(-)^{-1}:S\to S$ called inverse satisfying the following laws:

associativity: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ for all $a,b,c\in S$

left Malcev identity: $b \cdot b^{-1} \cdot a = a$ for all $a,b\in S$

right Malcev identity: $a \cdot b^{-1} \cdot b = a$ for all $a,b\in S$

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

Properties

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