Consider the following definition of a group:

Definition A semigroup S is said to be a group if the following hold:

1. There is an \(\displaystyle{e}\in{S}\) such that \(\displaystyle{e}{a}={a}\) for all \(\displaystyle{a}\in{S}\)

2. For each \(\displaystyle{a}\in{S}\) there is an element \(\displaystyle{a}^{{-{1}}}\in{S}\) with \(\displaystyle{a}^{{-{1}}}{a}={e}\)

At one point in my life, it seemed natural to ask what happens if we replace axiom 2 with the very similar axiom

2'. For each \(\displaystyle{a}\in{S}\) there is an element \(\displaystyle{a}^{{-{1}}}\in{S}\) with \(\displaystyle{a}{a}^{{-{1}}}={e}\).

It is a fun exercise to work out some of the consequences that result from this. Here are a few facts about a semigroup S which satisfies 1 and 2':

If e is the unique element of S satisfying axiom 1, then S is a group

If S has an identity (in the usual sense) then S is a group

The principal left ideal \(\displaystyle{S}{a}={\left\lbrace{s}{a}{\mid}{s}\in{S}\right\rbrace}\) is a group for all \(\displaystyle{a}\in{S}\), and in fact all principal left ideals of S are isomorphic as groups.

It is not difficult to find examples of such semigroups that are not groups. For example, consider the following set of \(\displaystyle{2}\times{2}\) matrices (with matrix multiplication as the operation):

\[\left\{(\begin{array}{c}a & b \\ 0 & 0 \end{array})| a,\ b \in \mathbb{R},\ a\ne0\right\}\]

Or, an example that appears as exercise 30 in section 4 of Fraleigh's abstract algebra text: the nonzero real numbers under the operation \(\displaystyle\times\) defined by \(\displaystyle{a}\times{b}={\left|{a}\right|}{b}\).

Certainly it is debatable whether or not semigroups satisfying axioms 1 and 2′ are "interesting" or "natural". But I guess I think they are. And, I am not the only one (or the first one, by a long shot!) to think this. See Mann, On certain systems which are almost groups (MR).