Group-subsets of monoids with different identities

Question

A subgroup must have the same identity as its containing group, but this fact requires inverses. I'm interested in subsets of monoids, which are groups in their own right, but vary greatly from the nature of the monoid itself, that is, it has a different identity, and the inverses are completely different as well (although we retain the same operation of course).

My favorite example of this is the set of $2 \times 2$ matrices in which every entry of the matrix is the same positive real number. This is a group, it is isomorphic to $\mathbb{R}_{>0}$, and yet the conventional identity matrix is not here, and none of these matrices are conventionally invertible. (Here, if $M$ denotes the matrix with $1$s everywhere then the identity matrix is $\frac12M$, and the isomorphism is $x\mapsto \frac{x}2M$.)

What are some other examples of this?

Answer 1

This is a well-studied area, related to the Green's relation $\mathcal H$. Let $M$ be a monoid. Define the relation $\mathcal H$ on $M$ by $$ a \mathrel{\mathcal H} b \iff aM = bM \text{ and }Ma = Mb $$ In other words, $a$ and $b$ generate the same right ideal and the same left ideal. Now, if $e$ is an idempotent, its $\mathcal H$-class is a subgroup of $M$ with identity $e$ and it is the largest subgroup of $M$ containing $e$.

In your case, the idempotent is the matrix $\pmatrix{1/2 &1/2\\1/2 &1/2}$.

Answer 2

Let $G$ be your favourite group. Consider the monoid $M_G$ with set $G\cup \{e\}$, where $e$ is a new letter acting as $eg=g=ge$ for all $g\in G$ (and let $G<M_G$ inherit the group operation from $G$).

Then we can view $G<M_G$ as a group, with a different identity element to $e\in M_G$.

For example, if you start with $G$ the trivial group, then $M_G$ can be viewed as $\{0, 1\}$ under multiplication inherited from $\mathbb{R}$ (here, $0$ corresponds to the trivial group, and $1=e$ is the added identity).

Answer 3

You could take a disjoint union of as many different groups as you like, together with an extra element $0$, and make all products between elements in different groups equal to $0$.