| It is known that the profinite completions of a free semigroup which are associated with a pseudovariety of semigroups or of ordered semigroups, can be defined by an écart or a quasi-écart. |
| In the case where the semigroup variety has a particular closure property with respect to programs, we are able to give precise characterizations of these regular languages. |
| The elements of the semigroup are identified with the reduced words of the rewriting system. |
| Equivalently a regular semigroup in which idempotents commute. |
| We also give some efficient algorithms to compute the Green relations, the local subsemigroups and the syntactic quasi-order of a subset of the semigroup. |
| Let S be a regular semigroup with set E of idempotent elements. |
