2025/09/18 by Jennifer Brown, Brown, Jennifer, Ricardo Suárez +1
Computer Science · Mathematics · #03E05 #06A05 #06A11 #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2509.14614
openalex publication_date 2025/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The countable condensation on a linear order L is the equivalence relation ∼ω defined by declaring x ∼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/ ∼ω ≅ 1. We define a multiplication operation ⋅ω on the class of linear orders by setting M ⋅ωL to be the order type of (ML)/ ∼ω (where ML denotes the lexicographic product), and show that the right identities for ⋅ω are exactly the uncountable suborders of U. The order types of these uncountable suborders of U form a left regular band under ⋅ω, and the order types of all suborders of U form a semigroup.