  We examine, in a general setting, a notion of inverse semigroup of left
quotients, which we call left I-quotients. This concept has appeared, and has
been used, as far back as Clifford's seminal work describing bisimple inverse
monoids in terms of their right unit subsemigroups. As a consequence of our
approach, we find a straightforward way of extending Clifford's work to
bisimple inverse semigroups (a step that has previously proved to be awkward).
We also put some earlier work on Gantos into a wider and clearer context, and
pave the way for further progress.
