nLab invertible semigroup

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.

1. Idea

An infertile semigroup is semigroup that is also an invertible magma.

2. Definition

With only multiplication

An invertible semigroup is a semigroup $(G,(-)\cdot(-) \colon G\times G\to G)$ 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 $(-)\cdot(-) \colon S\times S\to S$ 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 $a\in S$

3. Pseudo-torsor

Every invertible semigroup $G$ has a canonical structure of a pseudo-torsor, or associative Malcev algebra, via the ternary operation $t \colon G^3\to G$ defined as $t(x,y,z) = x\cdot y^{-1}\cdot z$. If the invertible semigroup is inhabited, then its associated pseudo-torsor is actually a heap, or equivalently a torsor.

4. Properties

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

5. Related concepts

• invertible magma (nonassociative version)

• semigroup (non-invertible version)

• group (unital version)

• commutative invertible semigroup (commutative version)

• possibly empty heap (forgetting the unit element)

• pseudo-torsor, torsor