Convergence and continuity in partially ordered sets and semilattices. (English) Zbl 0591.54029

Erné, Marcel; Gatzke, Hartmut

Continuous lattices and their applications, Proc. 3rd Conf., Bremen/Ger. 1982, Lect. Notes Pure Appl. Math. 101, 9-40 (1985).

Given a lattice S and a function t which assigns to each lattice a certain intrinsic lattice topology t(S), the fundamental problem is to determine when the lattice operations, considered as functions from \(S\times S\) into S are continuous. This problem has a relatively simple solution if \(S\times S\) has the topology t(S\(\times S)\), but in order to make t(S) a topological lattice it is necessary to assign to \(S\times S\) the product topology t(S)\(\times t(S)\). Thus the focus of the paper shifts from the original problem about topological lattices to a more basic question involving product invariance of intrinsic topologies: when is \(t(S\times S)=t(S)\times t(S)?\) This latter question is studied under fairly general assumptions about S and t. Nice results about continuity of lattice operations are given for an impressive assortment of intrinsic lattice topologies, including the Scott, Lawson, and order topologies, and a new characterization for continuous lattices (Corollary 4.6) is obtained. The concluding section examines the relationship between product invariance and the \(''T_ 2\)-ordered” property (that the order relation is closed in the product topology) for posets, complete lattices, and Boolean lattices.

Reviewer: D.C.Kent

MSC:

54H12	Topological lattices, etc. (topological aspects)
06B30	Topological lattices
06A12	Semilattices
54A20	Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54C05	Continuous maps
54B10	Product spaces in general topology

Keywords:

Scott topology; Lawson topology; continuous lattice; product invariance of intrinsic topologies; order topologies; complete lattices; Boolean lattices