A generalized direct product of Abelian groups
Let B be a countably complete Boolean algebra. The author introduces the concept of a quasi-sheaf (\({\mathcal S},\rho)\) over B, i.e. \({\mathcal S}(b)\) (b\(\in B)\) is a collection of Abelian groups and \(\rho^ b_ c: (b)\to (c)\) are homomorphisms satisfying certain restrictions, which are designed to guarantee that groups of the form \(\hat {\mathcal S}=(1)\) include Boolean powers. The latter groups have been discussed in preceding papers by the author [J. Algebra 82, 84-93 (1983; Zbl 0538.20027); Tsukuba J. Math. 6, 187-193 (1982; Zbl 0533.20026); Comment. Math. Univ. St. Pauli 32, 131-135 (1983; Zbl 0518.20054)]. Results from there are extended to the more general situation; among them: (a) If A is a slender group, then Hom(\(\hat {\mathcal S},A)\cong \otimes_{F\in {\mathcal F}}Hom(\hat {\mathcal S}/F,A)\), where \({\mathcal F}\) is the collection of all countably complete ultrafilters of B and \(\hat {\mathcal S}/F\) is the group \(\hat {\mathcal S}\) modulo \(K_ F^{{\mathcal S}}=\{x\in \hat {\mathcal S}|\) \(\exists b\in F\), \(\rho^ 1_ b(x)=0\}\). (b) If B is complete, G a countably complete filter of B, which is \(\aleph_ m\)- generated by a filter \(F\subseteq G\) \((\aleph_ m=first\) measurable cardinal), then \(K_ G^{{\mathcal S}}/K_ F^{{\mathcal S}}\) is a Fuchs-44- group. In contrast to (b) the final section deals with the algebraic compactness of filter quotients \(K_ G^{{\mathcal S}}/K_ F^{{\mathcal S}}\) in relation with two conditions on the filters \(F\subseteq G:\) \(''G\subseteq F^*(=least\) countably complete filter of B containing F) and ''F is pure in G''. This program has been carried out for \(B={\mathcal P}(I)\) by \textit{M. Dugas} and \textit{R. Göbel} [in J. Reine Angew. Math. 307/308, 341-352 (1979; Zbl 0396.20036); see also the reviewer, Lect. Notes Math. 874, 228-241 (1981; Zbl 0501.20034)]; their results remain valid in the more general context.
scientific article; zbMATH DE number 3886142
scientific article