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)?
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Pudlák, P., T⫲ma, J. Every finite lattice can be embedded in a finite partition lattice. Algebra Universalis 10, 74–95 (1980). https://doi.org/10.1007/BF02482893
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DOI: https://doi.org/10.1007/BF02482893