About the notion of semiring of sets. (Q1889542)

A nonempty class \(\mathcal P\) of subsets of a given set \(X\) is called a semiring if it has the following properties: (1) \(A\cap B\in {\mathcal P}\) whenever \(A,B\in {\mathcal P}\); (2) if \(A,B\in {\mathcal P}\) and \(A\subset B\) then there is a partition of \(B\setminus A\) into sets from \(\mathcal P\). In the paper under review the author introduces a more general version of those notion. Namely he replaces the condition (1) by the following: (\(1'\)) if \(A,B\in {\mathcal P}\) then there is a partition of \(A\cap B\) into sets from \(\mathcal P\). Next the author proves several properties of semirings according to the new definition. (Generally, properties of semirings in the old sense are preserved also for semirings in the wide sense.)