Accumulation points of congruence densities of finite lattices
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917334674833408 |
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| author | Czédli, Gábor |
| author_facet | Czédli, Gábor |
| contents | Let $\mathcal W$ be a nontrivial variety of lattices, and let $L$ be a finite lattice in $\mathcal W$. The congruence density of $L$ with respect to $\mathcal W$ is the number of congruences of $L$ divided by the maximum number of congruences of $|L|$-element lattices belonging to $\mathcal W$. We prove that, with respect to the order and multiplication of the real numbers, the set SCD$(\mathcal W)$ of congruence densities of finite members of $\mathcal W$ as well as its topological closure are countably infinite dually well-ordered monoids. We also prove that the set of accumulation points of SCD$(\mathcal W)$ is either a singleton or it is countably infinite; furthermore, it is a singleton if and only if $\mathcal W$ is a subvariety of the variety of modular lattices. This gives a complicated characterization of modularity: a non-singleton lattice $K$ is modular if and only if SCD$(\mathcal V(K))$, where $\mathcal V(K)$ denotes the variety generated by $K$, has only one accumulation point. The class $\mathcal S$ of semimodular lattices is not a variety, but SCD$(\mathcal S)$ is still meaningful; we prove that SCD$(\mathcal S)$ has exactly one accumulation point. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11454 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Accumulation points of congruence densities of finite lattices Czédli, Gábor Rings and Algebras 06B10, 06C10 Let $\mathcal W$ be a nontrivial variety of lattices, and let $L$ be a finite lattice in $\mathcal W$. The congruence density of $L$ with respect to $\mathcal W$ is the number of congruences of $L$ divided by the maximum number of congruences of $|L|$-element lattices belonging to $\mathcal W$. We prove that, with respect to the order and multiplication of the real numbers, the set SCD$(\mathcal W)$ of congruence densities of finite members of $\mathcal W$ as well as its topological closure are countably infinite dually well-ordered monoids. We also prove that the set of accumulation points of SCD$(\mathcal W)$ is either a singleton or it is countably infinite; furthermore, it is a singleton if and only if $\mathcal W$ is a subvariety of the variety of modular lattices. This gives a complicated characterization of modularity: a non-singleton lattice $K$ is modular if and only if SCD$(\mathcal V(K))$, where $\mathcal V(K)$ denotes the variety generated by $K$, has only one accumulation point. The class $\mathcal S$ of semimodular lattices is not a variety, but SCD$(\mathcal S)$ is still meaningful; we prove that SCD$(\mathcal S)$ has exactly one accumulation point. |
| title | Accumulation points of congruence densities of finite lattices |
| topic | Rings and Algebras 06B10, 06C10 |
| url | https://arxiv.org/abs/2603.11454 |