Sequential convergences in lattices. (English) Zbl 0763.06002

In the paper under review the notions of sequential convergence and multivalued convergence on a lattice \(L\) are introduced in a natural way. The introductory results describe some order properties of the partially ordered system of all convergences on \(L\) denoted by \(\text{Conv }L\) (e.g., the existence of the extremal elements, the construction of the join in \(\text{Conv }L\)). Analogous investigations concerning \(\ell\)- groups and Boolean algebras were published recently.
The author defines “positive” and “negative” convergences and by means of these concepts he characterizes \(\text{Conv }L\) like a direct product of two posets. The remaining parts of the paper are devoted to the study of convergences on linearly ordered sets and of intervals in \(\text{Conv }L\) in the case of distributive lattices.


06B30 Topological lattices
22A26 Topological semilattices, lattices and applications
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