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The
‘canonical embedding approach’ was introduced by the second
author and, subsequently, it has been applied several times to prove the
embeddability of certain regular extensions by groups into semidirect
products by groups. In the present paper this technique is generalized
so that it is suitable to handle regular extensions by inverse
semigroups. As an application, B. Billhardt's embedding theorem on
regular extensions of semilattices by inverse semigroups is
reproved.