A join-proximity lattice is a pair , where L is a bounded distributive lattice and ≺ is a binary relation on L satisfying certain conditions. This class of structures was introduced and studied by A. Jung and P. Sünderhauf in connection with stably compact spaces. In this paper, we present a spectral-type topological representation for the class of join-proximity lattices. We establish a categorical duality between join-proximity lattices and certain triples , where is a spectral space, and X_r is a subset of X satisfying additional conditions. As an application of this duality, we obtain a characterization of lattice congruences compatible with the relation ≺ in terms of specific subsets of prime filters, called cs-subsets.
In this paper, we prove a polynomial extension of Van der Waerden’s theorem near zero. We prove that if p_1, … , p_m ∈ℤ[x] are polynomials satisfying p_i(0) = 0 and if there exists δ > 0 such that p_i(x) > 0 for all x ∈ (0, δ ) and all i = 1, … , m , then for every sequence f: ℕ→ (0,1) satisfying ∑ _n ∈ℕ f(n) < 1 , and for every finite partition 𝒞 of (0, 1), there exist a cell C ∈𝒞 , an element a ∈ S , and a finite subset F ∈ P_f(ℕ) such that { a + p_i (∑ _t ∈ F f(t) ) : i = 1, 2, … , m }⊆ C.
We present a corrected version of a theorem from the paper "N-free posets and orthomodularity" published in Order 43(1) (2026).
The main aim of this paper is to present a Stone-type topological duality for posets, that is, we develop a categorical duality between the category of posets and a category of certain topological spaces. The principal tool to achieve this goal is the notion of ud-sets. More precisely, we introduce the notion of ud-sets, and then we use this notion to build a duality between posets without greatest elements and PSK-spaces, and a duality between posets with greatest elements and PSK^⊤ -spaces. Furthermore, we apply this dual equivalence to obtain Stone-type topological dualities for dcpos and complete lattices, respectively.
