Inverse Semigroup - The Basics

The Basics

The inverse of an element x of an inverse semigroup S is usually written x−1. Inverses in an inverse semigroup have many of the same properties as inverses in a group, for example, (ab)−1 = b−1a−1. In an inverse monoid, xx−1 and x−1x are not (necessarily) equal to the identity, but they are both idempotent. An inverse monoid S in which xx−1 = 1 = x−1x, for all x in S (a unipotent inverse monoid), is, of course, a group.

There are a number of equivalent characterisations of an inverse semigroup S:

Every element of S has a unique inverse, in the above sense.
Every element of S has at least one inverse (S is a regular semigroup) and idempotents commute (that is, the idempotents of S form a semilattice).
Every -class and every -class contains precisely one idempotent, where and are two of Green's relations.

The idempotent in the -class of s is s−1s, whilst the idempotent in the -class of s is ss−1. There is therefore a simple characterisation of Green's relations in an inverse semigroup:

$a\,\mathcal{L}\,b\Longleftrightarrow a^{-1}a=b^{-1}b,\quad a\,\mathcal{R}\,b\Longleftrightarrow aa^{-1}=bb^{-1}$