In this paper, by considering the notion of $Sigma$-hyperalgebras for an arbitrary signature $Sigma$, we study the notions of regular and strongly regular relations on a $Sigma$-hyperalgebra, $mathfrak{A}$. We show that each regular relation which contains a strongly regular relation is a strongly regular relation. Then we concentrate on the connection between the fundamental relation of $mathf...

Distance regular graphs fall into three families: primitive, antipodal, and bipar-tite. Each antipodal distance regular graph is a covering graph of a smaller (usually primitive) distance regular graph; the antipodal distance graphs of diameter three are covers of the complete graph, and are the first non-trivial case. Many of the known examples are connected with geometric objects, such as pro...

It is proved that the following conditions are equivalent: (a) there exists a complete, atomless, σ–centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra, (b) there exists a nowhere dense ultrafilter on ω. Therefore the existence of such algebras is undecidable in ZFC. In ”forcing language” condition (a) says that there exists a non– trivial σ–centered for...