Toby Bartels wrote:
OTOH, the version without unit probably has some use, just as non-monoid semigroups occasionally do.
Just a thought that this interest may be more than occasional: inverse semigroups are closely related to pseudogroups and so are part of Charles Ehresmann local-to-global view and interests, with a deep relation to ordered groupoids (see further work by Mark Lawson). So all these partial units express something on the *local* structure, even if there is a global unit. My paper with Aof (based on ideas of Pradines) uses an inverse semigroup generated by continuous local admissible sections. (with M. E.-S. A.-F. AOF), ``The holonomy groupoid of a locally topological groupoid'', {\em Top. and its Appl.}, 47 (1992) 97-113. Is this a rash view that one should look more at inverse categories? Ronnie Brown