The Radon-Nikodým property and the Kreĭn-Milman property are equivalent for strongly regular sets

Cycle cover property and CPP=SCC property are not equivalent

The regular element property

The property that an ideal whose annihilator is zero contains a regular element is examined from the point of view of constructive mathematics. It is shown that this property holds for nitely presented algebras over discrete elds, and for coherent, Noetherian, strongly discrete rings that contain an in nite eld. Let R be a commutative ring and M an R-module. For any subset I of R, we write AM(I...

The Ramanujan Property for Regular Cubical Complexes

We consider cubical complexes which are uniformized by an ordered product of regular trees. For these we define the notion of being Ramanujan, generalizing the one-dimensional definition introduced by Lubotzky, Phillips, and Sarnak [15]. As in [12], we also allow local systems. We discuss the significance of this property, and then we construct explicit arithmetic examples using quaternion alge...

Dilatation structures with the Radon-Nikodym property

The notion of a dilatation structure stemmed out from my efforts to understand basic results in sub-Riemannian geometry, especially the last section of the paper by Belläıche [2] and the intrinsic point of view of Gromov [5]. In these papers, as in other articles devoted to sub-Riemannian geometry, fundamental results admiting an intrinsic formulation were proved using differential geometry too...

The Moore and the Myhill Property For Strongly Irreducible Subshifts Of Finite Type Over Group Sets

We prove the Moore and the Myhill property for strongly irreducible subshifts over right amenable and finitely right generated left homogeneous spaces with finite stabilisers. Both properties together mean that the global transition function of each big-cellular automaton with finite set of states and finite neighbourhood over such a subshift is surjective if and only if it is preinjective. Thi...