Remarks on completeness of lattice-valued Cauchy spaces (Q2805907)

If \(L\) is a complete lattice, then a stratified \(L\)-filter on \(X\) is a mapping \(\mathcal F : L ^X \to L\) that preserves order, maps 0 to 0 and 1 to 1 and has the properties \(\mathcal F(a) \land \mathcal F(b) \leq \mathcal F(a \land b)\) for all \(a,b \in L^X\) and \(\alpha \leq \mathcal F(\alpha_X)\) for all \(\alpha \in L\) and \(\alpha_X\) being an \(\alpha\)-multiple of a characteristic function. The set of all stratified \(L\) filters is denoted by \(\mathcal F_L^s(X)\). A stratified \(L\)-Cauchy tower space is a universe with a set of mappings \(C_{\alpha} \subseteq \mathcal F_L^s(X)\) for all \(\alpha \in L\) fulfilling five conditions. Various definitions of stratified \(L\)-Cauchy tower spaces and their modifications are studied as well as completions of such spaces.