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Article overview
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Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices | | Gábor Czédli
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1 Jan 2023 | | Abstract: | Introduced by G. Gr"atzer and E. Knapp in 2007, a slim semimodular lattice
is a planar semimodular lattice without $M_3$ as a sublattice. We prove that if
$K$ is a slim semimodular lattice and $n$ denotes the number of its
join-irreducible congruence relations, then there exists a slim semimodular
lattice $L$ such that Con $L$ $cong$ Con $K$, the length of $L$ is at most
$2n^2$, and the number of elements of $L$ is at most $4n^4$. (In fact, we prove
slightly more.) Also, we present a new construction under which the class of
(isomorphism classes of) posets of join-irreducible congruences of slim
semimodular lattices is closed. | | Source: | arXiv, 2301.00401 | | Services: | Forum | Review | PDF | Favorites | |
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