Density and capacity of balleans generated by filters

Abstract

UDC 519.51

We consider a ballean $\mathbb B=(X,P,B)$ with an infinite support $X$ and a free filter $\phi$ on $X$ and define $B_{P\times\phi}(x,(\alpha,F))$ for every $\alpha\in P$ and $F\in \phi.$ The ballean $(X,P\times\phi, B_{P\times\phi})$ will be called the ballean-filter mix of $\mathbb B$ and $\phi$ and denoted by $\mathbb B(B,\phi).$ It was introduced in [O. V. Petrenko, I. V. Protasov, Balleans and filters, Mat. Stud., 38, No. 1, 3–11 (2012)] and was used to construction of a non-metrizable Frechet group ballean. In this paper some cardinal invariants are compared. In particular, we give a partial answer to the question: if we mix an ordinal unbounded ballean with a free filter of the subsets of its support, will the mix-structure's density be equal to its capacity, as it holds in the original balleans?