The countable condensation on linear orders
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915500888424448 |
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| author | Brown, Jennifer Suárez, Ricardo |
| author_facet | Brown, Jennifer Suárez, Ricardo |
| contents | The countable condensation on a linear order $L$ is the equivalence relation $\sim_ω$ defined by declaring $x \sim_ωy$ when the set of points between $x$ and $y$ is countable. We characterize the linear orders $L$ that condense to $1$ under the countable condensation by constructing a linear order $U$ that is universal for the order types $L$ such that $L/\!\!\sim_ω\, \cong 1$. We define a multiplication operation $\cdot_ω$ on the class of linear orders by setting $M \cdot_ωL$ to be the order type of $(ML)/\!\!\sim_ω$ (where $ML$ denotes the lexicographic product), and show that the right identities for $\cdot_ω$ are exactly the uncountable suborders of $U$. The order types of these uncountable suborders of $U$ form a left regular band under $\cdot_ω$, and the order types of all suborders of $U$ form a semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14614 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The countable condensation on linear orders Brown, Jennifer Suárez, Ricardo Logic 06A05, 06A11, 03E05 The countable condensation on a linear order $L$ is the equivalence relation $\sim_ω$ defined by declaring $x \sim_ωy$ when the set of points between $x$ and $y$ is countable. We characterize the linear orders $L$ that condense to $1$ under the countable condensation by constructing a linear order $U$ that is universal for the order types $L$ such that $L/\!\!\sim_ω\, \cong 1$. We define a multiplication operation $\cdot_ω$ on the class of linear orders by setting $M \cdot_ωL$ to be the order type of $(ML)/\!\!\sim_ω$ (where $ML$ denotes the lexicographic product), and show that the right identities for $\cdot_ω$ are exactly the uncountable suborders of $U$. The order types of these uncountable suborders of $U$ form a left regular band under $\cdot_ω$, and the order types of all suborders of $U$ form a semigroup. |
| title | The countable condensation on linear orders |
| topic | Logic 06A05, 06A11, 03E05 |
| url | https://arxiv.org/abs/2509.14614 |