The lattice theory of \(r\)-ordered partitions (Q1297483)

The paper deals with ordered partitions of sets considered from the lattice-theoretical viewpoint. Namely, the set \({\mathcal{P}}_r(\Omega)\) is the set of all \(r\)-ordered partitions of some set \(\Omega\), i.e. the sequences \(P=(P^{0},P^{1},\dots,P^{r-1})\) (\(r\in N,r\geq 2\)) of subsets of \(\Omega\) such that \(\bigcup P^{i} = \Omega\) and \(P^{i}\cap P^{j}=\emptyset\), for \(i\neq j\). The author defines an order relation \(\leq\) and operations \(\oplus,\otimes\) on \({\mathcal{P}}_r(\Omega)\) and proves that \({\mathcal{P}}_r(\Omega)\) is a complete Post algebra of order \(r\) (Cor. 2.8.2) and \(({\mathcal{P}}_r(\Omega),\oplus,\otimes)\) is a commutative ring with a unit of characteristic \(r\) (Th. 4.1).