Ladders and Squares
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918220306317312 |
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| author | Notaro, Lorenzo |
| author_facet | Notaro, Lorenzo |
| contents | In 1984, Ditor asked two questions: (1) For each $n\inω$ and infinite cardinal $κ$, is there a join-semilattice of breadth $n+1$ and cardinality $κ^{+n}$ whose principal ideals have cardinality $< κ$? (2) For each $n \in ω$, is there a lower-finite lattice of cardinality $\aleph_{n}$ whose elements have at most $n+1$ lower covers? We show that both questions have positive answers under the axiom of constructibility, and hence consistently with $\mathsf{ZFC}$. More specifically, we derive the positive answers from assuming that $\square_κ$ holds for enough $κ$'s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ladders and Squares Notaro, Lorenzo Logic Combinatorics 03E05 (Primary) 03E35, 06A07 (Secondary) In 1984, Ditor asked two questions: (1) For each $n\inω$ and infinite cardinal $κ$, is there a join-semilattice of breadth $n+1$ and cardinality $κ^{+n}$ whose principal ideals have cardinality $< κ$? (2) For each $n \in ω$, is there a lower-finite lattice of cardinality $\aleph_{n}$ whose elements have at most $n+1$ lower covers? We show that both questions have positive answers under the axiom of constructibility, and hence consistently with $\mathsf{ZFC}$. More specifically, we derive the positive answers from assuming that $\square_κ$ holds for enough $κ$'s. |
| title | Ladders and Squares |
| topic | Logic Combinatorics 03E05 (Primary) 03E35, 06A07 (Secondary) |
| url | https://arxiv.org/abs/2505.00414 |