algebra universalis

, Volume 10, Issue 1, pp 74–95

# Every finite lattice can be embedded in a finite partition lattice

• Pavel Pudlák
• Jiří T⫲ma
Article

## 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)?

## Keywords

Triangle Inequality Algebra UNIV Congruence Lattice Boolean Lattice Finite Lattice
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Pavel Pudlák
• 1
• 2
• Jiří T⫲ma
• 1
• 2
1. 1.Mathematics InstitutCzechoslovak Academy of SciencePrahaCzechoslovakia
2. 2.Nuclear Physics FacultyČVUTPrahaCzechoslovakia