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

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 |