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Cevian operations on distributive lattices

Abstract : We construct a completely normal bounded distributive lattice D in which for every pair (a, b) of elements, the set {x ∈ D | a ≤ b ∨ x} has a countable coinitial subset, such that D does not carry any binary operation - satisfying the identities x ≤ y ∨(x-y), (x-y)∧(y-x) = 0, and x-z ≤ (x-y)∨(y-z). In particular, D is not a homomorphic image of the lattice of all finitely generated convex ℓ-subgroups of any (not necessarily Abelian) ℓ-group. It has ℵ 2 elements. This solves negatively a few problems stated by Iberkleid, Martínez, and McGovern in 2011 and recently by the author. This work also serves as preparation for a forthcoming paper in which we prove that for any infinite cardinal λ, the class of Stone duals of spectra of all Abelian ℓ-groups with order-unit is not closed under L ∞λ-elementary equivalence.
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Contributor : Friedrich Wehrung <>
Submitted on : Friday, September 6, 2019 - 12:03:20 PM
Last modification on : Monday, April 27, 2020 - 4:14:03 PM

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Friedrich Wehrung. Cevian operations on distributive lattices. Journal of Pure and Applied Algebra, Elsevier, 2020, 224 (4), pp.106202. ⟨10.1016/j.jpaa.2019.106202⟩. ⟨hal-01988169v3⟩

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