ON EXISTENCE VARIETIES OF LOCALLY INVERSE-SEMIGROUPS

A locally inverse semigroup is a regular semigroup S with the property that eSe is inverse for each idempotent e of S. Motivated by natural examples such as inverse semigroups and completely simple semigroups, these semigroups have been the subject of deep structure-theoretic inv estigations. The class LJ of locally inverse semigroups forms an exist ence variety (or e-variety): a class of regular semigroups closed unde r direct products, homomorphic images and regular subsemigroups. We co nsider the lattice L(LJ) of e-varieties of such semigroups. In particu lar we investigate the operations of taking meet and join with the e-v ariety CL of completely simple semigroups. An important consequence of our results is a determination of the join of CL with the e-variety o f inverse semigroups - it comprises the E-solid locally inverse semigr oups. It is shown, however, that not every e-variety of E-solid locall y inverse semigroups is the join of completely simple and inverse e-va rieties.