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Atomicity of the Boolean algebra of direct factors of a directed set. (English) Zbl 0938.06011
Let $$L$$ and $$L_i$$ $$(i\in I)$$ be directed sets. If $$\varphi$$ an isomorphism of $$L$$ onto the direct product $$\prod _{i\in I} L_i$$, then the relation $$\varphi : L \rightarrow \prod _{i\in I} L_i$$ is called a direct product decomposition of $$L$$. An internal direct product decomposition of $$L$$ is defined in a corresponding way. A refinement theorem is stated in 2.7 as follows: Let $$L=\prod _{i\in I} A_i$$ and $$L=\prod _{j\in J} B_j$$ be two internal direct decompositions of $$L$$. Then $$L=\prod _{i\in I, j\in J} (A_i\cap B_i)$$ is a common refinement of both decompositions. Namely, for each $$i\in I$$ and each $$j\in J$$ we have $$A_i =\prod _{j\in J} (A_i \cap B_j)$$ and $$B_j = \prod _{i\in I}(A_i \cap B_j)$$. Let $$D(L)$$ be defined as the partially ordered set consisting of all internal direct factors of $$L$$. An extensive study of $$D(L)$$ and of direct decompositions of intervals in $$L$$ is carried out in Sections 3 and 4 of the paper. The result of 3.14 deserves to be picked out as a sample statement: The partially ordered set $$D(L)$$ is a Boolean algebra.
Reviewer: L.Beran (Praha)

##### MSC:
 600000 Structure theory of Boolean algebras
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