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arXiv · 2410.15328

Four generators of an equivalence lattice with consecutive block counts

Gábor Czédli
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Abstract

The block count of an equivalence $\mu\in$ Equ$(A)$ is the number blnum$(\mu)$ of blocks of (the partition corresponding to) $\mu$. We say that $X=\{\mu_1,\mu_2,\mu_3,\mu_4\}$ is a four-element generating set of Equ$(A)$ with consecutive block counts if $X$ generates Equ$(A)$ and blnum$(\mu_{i+1})$ = blnum$(\mu_{1})+i$ for $i\in\{1,2,3\}$. We prove that if the number of elements of a finite set $A$ is six or at least eight, then Equ$(A)$ has a four-element generating set with consecutive block counts. Also, we present a historical remark on the connection between equivalence lattices and quasiorder lattices.

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Gábor Czédli. 2024-10-20. Four generators of an equivalence lattice with consecutive block counts. https://arxiv.org/abs/2410.15328

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