Atomicity of the Boolean algebra of direct factors of a directed set.

*(English)*Zbl 0938.06011Let \(L\) 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:

06E05 | Structure theory of Boolean algebras |