Every finite lattice can be embedded in a finite partition lattice
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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 . 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)?
KeywordsTriangle Inequality Algebra UNIV Congruence Lattice Boolean Lattice Finite Lattice
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