Abstract We present some generalizations of the well-known correspondence, found by Exel, between partial actions of a group G on a set X and semigroup homomorphism of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒮</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{\mathcal{S}}(G)} on the semigroup <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>I</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {I(X)} of partial bijections of X , with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒮</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{\mathcal{S}}(G)} being an inverse monoid introduced by Exel. We show that any unital premorphism <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>θ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>→</m:mo> <m:mi>S</m:mi> </m:mrow> </m:mrow> </m:math> {\theta:G\to S} , where S is an inverse monoid, can be extended to a semigroup homomorphism <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>θ</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>:</m:mo> <m:mrow> <m:mi>T</m:mi> <m:mo>→</m:mo> <m:mi>S</m:mi> </m:mrow> </m:mrow> </m:math> {\theta^{*}:T\to S} for any inverse semigroup T with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="script">𝒮</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊆</m:mo> <m:mi>T</m:mi> <m:mo>⊆</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>P</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>×</m:mo> <m:mi>G</m:mi> </m:mrow> </m:mrow> </m:math> {\operatorname{\mathcal{S}}(G)\subseteq T\subseteq P^{*}(G)\times G} , with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>P</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {P^{*}(G)} being the semigroup of non-empty subsets of G , and such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>E</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {E(S)} satisfies some lattice-theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\Gamma(X)} , the inverse semigroup of partial homeomorphisms between open subsets of a locally compact Hausdorff space X .
