Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991 Association of Mizar Users

## Filters - Part II. Quotient Lattices Modulo Filters and Direct Product of Two Lattices

Grzegorz Bancerek
Warsaw University, Bialystok

### Summary.

Binary and unary operation preserving binary relations and quotients of those operations modulo equivalence relations are introduced. It is shown that the quotients inherit some important properties (commutativity, associativity, distributivity, etc.). Based on it, the quotient (also called factor) lattice modulo a filter (i.e. modulo the equivalence relation w.r.t the filter) is introduced. Similarly, some properties of the direct product of two binary (unary) operations are present and then the direct product of two lattices is introduced. Besides, the heredity of distributivity, modularity, completeness, etc., for the product of lattices is also shown. Finally, the concept of isomorphic lattices is introduced, and there is shown that every Boolean lattice \$B\$ is isomorphic with the direct product of the factor lattice \$B/[a]\$ and the lattice latt\$[a]\$, where \$a\$ is an element of \$B\$.

#### MML Identifier: FILTER_1

The terminology and notation used in this paper have been introduced in the following articles              

Contents (PDF format)

#### Bibliography

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