2021/12/02 by Martínez, Luis, Pinedo, Héctor, Uzcátegui, Carlos
#Dynamical Systems (math.DS) #FOS: Mathematics #General Topology (math.GN) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2112.01289
We present some generalizations of the well-known correspondence, found by R. Exel, between partial actions of a group G on a set X and semigroup homomorphism of S(G) on the semigroup I(X) of partial bijections of X, being S(G) an inverse monoid introduced by Exel. We show that any unital premorphism θ:G→ S, where S is an inverse monoid, can be extended to a semigroup homomorphism θ^*:T→ S for any inverse semigroup T with S(G)⊆ T⊆ P^*(G)× G, being P^*(G) the semigroup of non-empty subset of G, and such that 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 Γ(X), the inverse semigroup of partial homeomorphism between open subsets of a locally compact Hausdorff space X.