Regular non-orthodox semigroups Is there a resource somewhere with smallest finite and other examples of regular semigroups that are not orthodox? I want concrete examples for my private research. I once installed GAP and two semigroup packages too but I have no idea how difficult it would be to calculate such examples there, I haven't used GAP before.
 A: There is only one finite semigroup of order 4 whose idempotents don't form a subsemigroup (up to anti-isomorphism), and has the following Cayley table:
\begin{array}{l|llll}
  & 1 & 2 & 3 & 4 \\ \hline
1 & 1 & 1 & 1 & 1 \\
2 & 1 & 1 & 1 & 2 \\
3 & 1 & 2 & 3 & 2 \\
4 & 1 & 1 & 1 & 4
\end{array}
Sadly it is not regular. It has 3 idempotents and one non-regular element (the 2).
However it might be possible to extend this semigroup by adding another element that should become an inverse of 2:
\begin{array}{l|lllll}
  & 1 & 2 & 3 & 4 & 5 \\ \hline
1 & 1 & 1 & 1 & 1 &   \\
2 & 1 & 1 & 1 & 2 & 3 \\
3 & 1 & 2 & 3 & 2 &   \\
4 & 1 & 1 & 1 & 4 & 5 \\
5 &   & 4 & 5 &   &  
\end{array}
As I was writing this, I did a quick program and indeed this is possible: there is only one way to extend it and the resulting regular non-orthodox semigroup is this one:
\begin{array}{l|lllll}
  & 1 & 2 & 3 & 4 & 5 \\ \hline
1 & 1 & 1 & 1 & 1 & 1 \\
2 & 1 & 1 & 1 & 2 & 3 \\
3 & 1 & 2 & 3 & 2 & 3 \\
4 & 1 & 1 & 1 & 4 & 5 \\
5 & 1 & 4 & 5 & 4 & 5
\end{array}
A: Hint. Just apply the definition. An orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup. Thus you are looking for a regular semigroup in which there are two idempotents whose product is not idempotent. You should be able to find a 5-element regular semigroup with this property.
Edit. See this answer for a complete description of the minimal counterexample.
