Every finite lattice can be embedded in a finite partition lattice |
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Authors: | Pavel Pudlák Jiří T⫲ma |
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Institution: | (1) Mathematics Institut, Czechoslovak Academy of Science, Praha, Czechoslovakia;(2) Nuclear Physics Faculty, ČVUT, Praha, Czechoslovakia |
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Abstract: | Conclusions There are many questions, which arise in connection with the theorem presented. In general, we would like to know more about
the class of embeddings of a given lattice in the lattices of all equivalences over finite sets. Some of these problems are
studied in 4]. In this paper, an embedding is called normal, if it preserves 0 and 1. Using regraphs, our result can be easily
improved as follows:
THEOREM.For every lattice L, there exists a positive integer n
0,such that for every n≥n
0,there is a normal embedding π: L→Eq(A), where |A|=n.
Embedding satisfying special properties are shown in Lemma 3.2 and Basic Lemma 6.2. We hope that our method of regraph powers
will produce other interesting results.
There is also a question about the effectiveness of finding an embedding of a given lattice. In particular, the proof presented
here cannot be directly used to solve the following.
Problem. Can the dual of Eq(4) be embedded into Eq(21000)?
Presented by G. Gr?tzer. |
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Keywords: | |
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