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 investigations. The class LJ of locally inverse semigroups forms an existence variety (or e-variety): a class of regular semigroups closed under direct products, homomorphic images and regular subsemigroups. We consider the lattice L(LJ) of e-varieties of such semigroups. In particular we investigate the operations of taking meet and join with the e-variety CL of completely simple semigroups. An important consequence of our results is a determination of the join of CL with the e-variety of inverse semigroups - it comprises the E-solid locally inverse semigroups. It is shown, however, that not every e-variety of E-solid locally inverse semigroups is the join of completely simple and inverse e-varieties.