Cross-connection structure of locally inverse semigroups

Locally inverse semigroups are regular semigroups whose idempotents form pseudosemilattices. We characterize the categories that correspond to locally inverse semi groups in the realm of Nambooripad's cross-connection theory. Further, we specialize our cross-connection description of locally inverse semigroups to inverse semigroups and completely 0-simple semigroups, obtaining structure theorems for these classes. In particular, we show that the structure theorem for inverse semigroups can be obtained using only one category, quite analogous to the Ehresmann-Schein-Nambooripad Theorem; for completely 0-simple semigroups, we show that cross-connections coincide with structure matrices, thus recovering the Rees Theorem by categorical tools.