On lattices of uniformities (Q2476637)

The author studies lattice-theoretical properties of lattices of uniformities, such as modularity, distributive laws and the existence of (relative) complements. In these investigations the concept of permutable uniformities turns out to be useful: Two uniformities \(u,v\) on a set \(X\) are said to be permutable if \(v\circ u = u\circ v\) where \(u\circ v\) denotes the filter generated by the filter base \(\{U\circ V: U\in u,V\in v\}\). Also, independent families of uniformities play an important role, where a family of uniformities on a set \(X\) is called independent if the induced topologies are independent. A family \((\tau_\alpha)_{\alpha\in A}\) of topologies on a set \(X\) is said to be independent if \(\bigcap_{\alpha\in F}O_\alpha\not =\emptyset\) whenever \(O_\alpha\) is a nonempty open set in \((X,\tau_\alpha)\) and \(F\) is a finite subset of \(A.\) The author mainly concentrates on lattice uniformities on a lattice \(L\), that is, those uniformities on \(L\) such that the lattice operations \(\vee\) and \(\wedge\) are uniformly continuous. Among other things it is shown that the lattice of all lattice uniformities on a lattice \(L\) is a closed sublattice of the lattice of all uniformities on \(L\). (Here a subset \(M\) of a complete lattice \(L\) containing both \(\inf Z\) and \(\sup Z\) whenever \(Z\subseteq M\) is called a closed sublattice of \(L\).)