Ladders and Squares

2025/05/01 by Notaro, Lorenzo
#03E05 (Primary) 03E35 #06A07 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2505.00414

Abstract

In 1984, Ditor asked two questions: (1) For each n∈ω 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 ∈ ω, is there a lower-finite lattice of cardinality ℵ_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 ZFC. More specifically, we derive the positive answers from assuming that \square_κ holds for enough κ's.

Citations

• Lattices of regular closed subsets of closure spaces
• On the consistency strength of the proper forcing axiom
• Infinite combinatorial issues raised by lifting problems in universal algebra
• Large semilattices of breadth three
• Distributive congruence lattices of congruence-permutable algebras
• Representation of algebraic distributive lattices with ℵ_1 compact elements as ideal lattices of regular rings

Related

• Sublattices of complete lattices with continuity conditions
• Degrees of join-distributivity via Bruns-Lakser towers
• A characterization of well-founded algebraic lattices
• Join-semilattices whose principal filters are pseudocomplemented lattices
• Some semilattices of definable sets in continuous logic