nLab invertible semigroup

This entry is about semigroups with a two-sided inverse. For the semigroup 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 inverse semigroup.

group theory

Cohomology and Extensions

Representation theory

representation theory

geometric representation theory

Contents

Idea

A semigroup that is also an invertible magma.

Definition

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 $a,b\in S$
• commutativity with inverse elements: $a \cdot a^{-1} = a^{-1} \cdot a$ for all $a\in S$

Pseudo-torsor

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.