Topologies of Lattice Products

Abstract

A number of different ways of defining topologies in a lattice or partially ordered set in terms of the order relation are known. Three of these methods have proved to be useful and convenient for lattices of special types, namely theidealtopology, theintervaltopology, and thenew intervaltopology of Garrett Birkhoff. In another paper (2) we have shown that these three topologies are equivalent for chains (totally ordered sets), where they reduce to the usual intrinsic topology of the chain.Since many important lattices are either direct products of chains or sublattices of such products, it is natural to ask what relationships exist between the various order topologies of a direct product of lattices and those of the lattices themselves.