Directed Sets, Nets, Ideals, Filters, and Maps

Grzegorz Bancerek

Institute of Mathematics, Polish Academy of Sciences

Summary.

Notation and facts necessary to start with the formalization of continuous lattices according to [8] are introduced. The article contains among other things, the definition of directed and filtered subsets of a poset (see 1.1 in [8, p. 2]), the definition of nets on the poset (see 1.2 in [8, p. 2]), the definition of ideals and filters and the definition of maps preserving arbitrary and directed sups and arbitrary and filtered infs (1.9 also in [8, p. 4]). The concepts of semilattices, sup-semiletices and poset lattices (1.8 in [8, p. 4]) are also introduced. A number of facts concerning the above notion and including remarks 1.4, 1.5, and 1.10 from [8, pp. 3-5] is presented.

This work has been partially supported by Office of Naval Research Grant N00014-95-1-1336.

MML Identifier: WAYBEL_0

Contents (PDF format)

1. Directed subsets
2. Nets
3. Lower and upper subsets
4. Ideals and filters
5. Chains
6. Semilattices
7. Maps
8. Completeness wrt directed sets

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