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Further, we specialize our cross-connection description of locally inverse semigroups to inverse semigroups and completely [Formula: see text]-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\u2013Schein\u2013Nambooripad Theorem; for completely [Formula: see text]-simple semigroups, we show that cross-connections coincide with structure matrices, thus recovering the Rees Theorem by categorical tools.<\/jats:p>","DOI":"10.1142\/s0218196723500091","type":"journal-article","created":{"date-parts":[[2022,10,28]],"date-time":"2022-10-28T03:45:07Z","timestamp":1666928707000},"page":"123-159","source":"Crossref","is-referenced-by-count":2,"title":["Cross-connection structure of locally inverse semigroups"],"prefix":"10.1142","volume":"33","author":[{"given":"P. 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