Limits in the category of topological molecular lattices (Q674663)

From an analogous result proved by the author for molecular lattices the author derives the following theorem (The limit in \textbf{TML}, the category of the topological molecular lattices): Let \(J\) be a small category, \(F:J\rightarrow\mathbf{TML}\) be a functor, for all \(j\in J,F(j)=(L_j,\delta _j)\) be topological molecular lattices and for all \(J\)-morphisms \(u:i\rightarrow j\), \( F(u):F(i)\rightarrow F(j)\) be continuous generalized order-morphisms. Then the limit of \(F\) is \(((R_{\prec *}(L),\delta ),(\pi _j)_{j\in J}) \), where \(L=\{f\in \prod_{j\in J}L_j|\) for any \(J\)-morphism \(u:i\rightarrow j,f(j)=F(u)\cdot f(i)\}\), and for all \(f,g\in L,f\leq g\) iff for all \(j\in J,f(j)\leq g(j)\) in \( L_j,f\prec g\) iff for all \(j\in J,f(j) ? g(j)\) in \(L_j\); for any \(j\in J\), \(\pi _j:R_{\prec *}(L)\rightarrow L_j\) is defined by \(\pi _j(I)=\sup \cdot P_j(I)\) for \(I\in R_{\prec *}(L),\) provided that for all \(f\in L,P_j(f)=f(j)\); \(\delta \) is the coarser cotopology on \(R_{\prec *}(L)\) such that every \(\pi _j(j\in J) \) is continuous. Here if \(L\) is a complete lattice and \(x,y\in L\) we say that \(x ? y\) if for every subset \(A\) of \(L\) with \(\sup A\geq y\) there is a \(d\in A\) such that \(d\geq x.\) Particular cases: product, (multiple) equalizer are derived. Finally also the inverse limit in \textbf{TML }is characterized\textbf{.} The reader is referred to ``Theory of topological molecular lattices'' by \textit{Wang Guojun} [Fuzzy Sets Syst. 47, No. 3, 351-376 (1992; Zbl 0783.54032)] for the concept of topological molecular lattice.