Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996 Association of Mizar Users

## 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

The terminology and notation used in this paper have been introduced in the following articles [12] [6] [14] [15] [17] [16] [7] [4] [5] [11] [3] [10] [1] [2] [13] [9]

#### 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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