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Extensions by inverse semigroups andλ -semidirect products

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.