Topologies on products of partially ordered sets III: Order convergence and order topology

Abstract

This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are: (1) Order convergence in a product of posets is obtained componentwise if and only if the number of non-bounded posets occurring in this product is finite (1.5). (2) For any product of posets, the projections are open and continuous with respect to the order topologies (2.1). (3) A productL of chainsLi has topological order convergence iff all but a finite number of the chains are bounded. In this case, the order topology onL agrees with the product topology (2.7). (4) If (Li:j ∈J) is a countable family of lattices with topological order convergence and first countable order topologies then order topology of the product lattice and product topology coincide (2.8). (5) LetP1 be a poset with topological order convergence and locally compact order topology. Then for any posetP2, the order topology ofP1⊗P2 coincides with the product topology (2.10). (6) A latticeL which is a topological lattice in its order topology is join- and meet-continuous. The converse holds whenever the order topology ofL⊗L is the product topology (2.15).