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.

**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.