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

Contents (PDF format)

Bibliography

[1] Grzegorz Bancerek. The well ordering relations. Journal of Formalized Mathematics, 1, 1989.

[2] Grzegorz Bancerek. Filters --- part I. Journal of Formalized Mathematics, 2, 1990.

[3] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989.

[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.

[6] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.

[7] Czeslaw Bylinski. The modification of a function by a function and the iteration of the composition of a function. Journal of Formalized Mathematics, 2, 1990.

[8] Konrad Raczkowski and Pawel Sadowski. Equivalence relations and classes of abstraction. Journal of Formalized Mathematics, 1, 1989.

[9] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.

[10] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[11] Andrzej Trybulec. Finite join and finite meet, and dual lattices. Journal of Formalized Mathematics, 2, 1990.

[12] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[13] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[14] Stanislaw Zukowski. Introduction to lattice theory. Journal of Formalized Mathematics, 1, 1989.