Abstract
We show how the correspondence between inverse semigroups and inductive groupoids (a class of ordered groupoids) leads to a reinterpretation of McAlister's P-Theorem as a function extension theorem. This result can only be expressed, in general, by working within a larger category: namely that of functorially ordered groupoids. The consequences of this approach for the theory of E-unitary covers of inverse semigroups and its generalisations are worked out in subsequent papers.
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