We study algebraic and topological properties of the convolution semigroups of probability measures on a topological groups and show that a compact Clifford topological semigroup ▫$S$▫ embeds into the convolution semigroup ▫$P(G)$▫ over some topological group ▫$G$▫ if and only if ▫$S$▫ embeds into the semigroup ▫$\exp(G)$▫ of compact subsets of ▫$G$▫ if and only if ▫$S$▫ is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup ▫$S$▫ embeds into the functor-semigroup ▫$F(G)$▫ over a suitable compact topological group ▫$G$▫ for each weakly normal monadic functor ▫$F$▫ in the category of compacta such that ▫$F(G)$▫ contains a ▫$G$▫-invariant element (which is an analogue of the Haar measure on ▫$G$▫).
COBISS.SI-ID: 15950681
The aim of this work is to characterize the discrete gradient vector fields on infinite and locally finite simplicial complexes which are induced by a proper discrete Morse function. This characterization is essentially given by the non-existence of closed trajectories and the absence of a certain kind of incidence between monotonous rays in the given field.
COBISS.SI-ID: 15865945
We show that the dimension of the sublinear Higson corona of a metric space ▫$X$▫ is the smallest non-negative integer ▫$m$▫ with the following property: Any norm-preserving asymptotically Lipschitz function from a closed subset ▫$A$▫ of ▫$X$▫ to the Euclidean space of dimension ▫$m+1$▫ extends to a norm-preserving asymptotically Lipschitz function from ▫$X$▫ to the Euclidean space of dimension ▫$m+1$▫. As an application we obtain another proof of the following result of Dranishnikov and Smith: Let ▫$X$▫ be a cocompact proper metric space, which is ▫$M$▫-connected for some $M$, and has the asymptotic Assouad-Nagata dimension finite. Then this dimension equals the dimension of the sublinear Higson corona of ▫$X$▫.
COBISS.SI-ID: 16135001