Regular non-orthodox semigroups

Question

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.

Answer 1

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.

Answer 2

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}