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.
We show that for any finite lattice L of order dimension d , the global dimension of its incidence algebra is at most d . We also provide an explicit family of finite posets demonstrating that this inequality does not hold in general once the lattice assumption is removed.
The minimal prime elements of algebraic frames with the finite intersection property (FIP) can be characterized as those prime elements which are the joins of the pseudo-complements of all compact elements not below the said prime. Our goal is to study the class of algebraic frames with this latter property, generalizing the FIP case.
In this paper, we algebraically study the {∨ ,→ ,□ , , ,⊤} -fragments of the Positive Intuitionistic Modal Logic (i.e., the Positive Modal Logic PML with an intuitionistic implication), as well as those of the intuitionistic modal logic FS defined by G. Fischer Servi. We introduce and define the varieties of Positive Modal Hilbert algebras (PMHil -algebras) and Fischer Servi Modal Hilbert algebras (FSMHil -algebras) and we establish spectral-like dualities for these algebras. We define the categories 𝖯𝖲𝖾𝗆𝖧𝗂𝗅 and 𝖥𝖲𝖲𝖾𝗆𝖧𝗂𝗅 , whose objects are PMHil -algebras and FSMHil -algebras, respectively, and whose morphisms are □ -semi-homomorphisms. By considering □ -homomorphisms instead, we obtain the categories 𝖯𝖧𝗈𝗆𝖧𝗂𝗅 and 𝖥𝖲𝖧𝗈𝗆𝖧𝗂𝗅, respectively. Furthermore, we prove that these categories are dually equivalent to categories of H_0^∨ -spaces endowed with a special binary relation and certain special continuous maps. The established dualities enable us to characterize the congruences in PMHil -algebras and FSMHil -algebras.
NFTs or non-fungible tokens are digital assets stored on a blockchain. They can be traded or exchanged for money, cryptocurrencies or other NFTs. Examples include works of art and digital or other tokenised collectables. An important determinant of price for collectables is rarity within a collection. Many trading platforms offer to rank items in terms of rarity but rankings differ considerably and, often, little explanation is given of the methods used. This paper provides a mathematical framework for the analysis of a comprehensive class of collections. It examines individual and joint distributions of attributes over such collections, and shows how these can be combined to provide a rarity ranking for all items in the collection. There is, however, only a limited range of methods that give consistent results over different collections. These are identified as belonging to a one-parameter family of ranking functions. Each gives to every item of a collection a rarity score that is directly comparable between collections. Despite taking account of all possible combinations of attributes when ranking, the method is nonetheless computationally feasible.
In this paper, we study the simplex faces of the order polytope 𝒪(P) and the chain polytope 𝒞(P) of a finite poset P. We show that, if P can be recursively constructed from X -free posets using disjoint unions and ordinal sums, then 𝒞(P) has at least as many k-dimensional simplex faces as 𝒪(P) does, for each dimension k. This generalizes a previous result of Mori, both in terms of the dimensions of the simplices and in terms of the class of posets considered.
Given a locale L , the ordered collection $$\textsf{S}_c(L)$$ S c ( L ) of joins of closed sublocales forms a frame—somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of L , where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc. A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether $$\textsf{S}_c(L)$$ S c ( L ) is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale L such that $$\textsf{S}_c(L)$$ S c ( L ) is not a coframe. The main challenge in such question lies in the difficulty of understanding exact infima in $$\textsf{S}_c(L)$$ S c ( L ) ; we circumvent this by analysing a certain separation property satisfied by $$\textsf{S}_c(L)$$ S c ( L ) .
A family 𝒢 of sets is a(n induced) copy of a poset P=(P,⩽) if there exists a bijection b:P→𝒢 such that p⩽q holds if and only if b(p)⊂b(q) . The induced saturation number sat^*(n,P) is the minimum size of a family ℱ⊆2^[n] that does not contain any copy of P , but for any G∈2^[n]∖ℱ , the family ℱ∪{G} contains a copy of P . We consider sat^*(n,P) for posets P that are formed by pairwise incomparable chains, i.e. P=⊕_j=1^mC_i_j . We make the following two conjectures: (i) sat^*(n,P)=O(n) for all such posets and (ii) sat^*(n,P)=O(1) if not all two chains are of the same size. (The second conjecture is known to hold if there is a unique longest among the chains.) We verify these conjectures in some special cases: we prove (i) if all chains are of the same length, we prove (ii) in the first unknown general case: for posets 2C_k+C_1 . Finally, we give an infinite number of examples showing that (ii) is not a necessary condition for sat^*(n,P)=O(1) among posets P=⊕_j=1^mC_i_j : we prove sat^*(n,(( [ 2t; t ]) +1)C_2)=O(1) for all t≥2 .
The finite condensation ∼ _F is an equivalence relation defined on a linear order L by x ∼ _F y if and only if the set of points lying between x and y is finite. We define an operation · _F on linear orders L and M by L · _F M = o.t.( (LM)/∼ _F) ; that is, L · _F M is the order type of the lexicographic product of L and M modulo the finite condensation. The infinite order types L such that L / ∼ _F ≅ 1 are ω , ω ^*, and ζ (where ω ^* is the reverse ordering of ω , and ζ is the order type of ℤ ). We show that under the operation · _F , the set R={1, ω , ω ^*, ζ} forms a left regular band. Further, each of the ordinal elements of R defines, via left or right multiplication modulo the finite condensation, a weakly order-preserving map on the class of ordinals. We study these maps’ effect on the ordinals of finite degree in Cantor normal form. In particular, we examine the extent to which one of these maps, sending α to 1 · _Fα≅o.t. (^α/_∼ _F ) , behaves similarly to a derivative operator on the ordinals of finite degree in Cantor normal form.
We extend the work of Galatos (2004) on nested sums, originally called generalised ordinal sums, of residuated lattices. We show that the nested sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a recent paper by Craig and Robinson (2025) the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their nested sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable.
We say that two partial orders on [n] are compatible if there exists a partial order that refines both of them. This compatibility relation induces a natural set system structure between the collection ℱ of all partial orders and the collection 𝒢 of all total orders on [n] , where each order is identified with the set of orders compatible with it. In this note, we determine the VC-dimension of ℱ with respect to 𝒢 , proving that VC_𝒢(ℱ) = ⌊n^2/4⌋ for n ⩾ 4 . We also establish bounds on the dual VC-dimension, showing that 2(n-3)⩽VC_ℱ(𝒢) ⩽ n log _2 n for all n ⩾ 1 .
The class of semi-boolean ℓ -groups was introduced in 1968 by A. Bigard. These are the ℓ -groups G in which the principal convex ℓ -subgroup G(a) generated by any a ∈ G is equal to the polar a^⊥⊥ . Examples include all hyperarchimedean ℓ -groups and all existentially closed abelian ℓ -groups. Ordered by inclusion, the set of convex ℓ -subgroups of a semi-boolean ℓ -group is a Martínez frame (an algebraic frame with FIP in which every element is a d-element). Related are the Yosida ℓ -groups, i.e., the ℓ -groups whose frame of convex ℓ -subgroups is a Yosida frame (an algebraic frame with FIP in which every compact element is a meet of maximal elements). Applying results on Martínez frames and Yosida frames, we obtain new characterizations of the semi-boolean and Yosida ℓ -groups, show that the former constitute a radical class and the latter do not, and present new examples with special properties. To build some of our examples, we introduce the G+B construction for ℓ -groups, an adaptation of the A+B construction from commutative algebra.
The FKG inequality is a powerful tool for proving inequalities in distributive lattices. We show how a special case, which we call the Order Ideal Lemma, can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner. We use the Order Ideal Lemma to prove log-concavity and log-convexity of various sequences involving lattice paths (Catalan, Motzkin and large Schröder numbers), intervals in Young’s lattice, order polynomials, specializations of Schur and Schur Q-functions, Lucas sequences, descent and peak polynomials of permutations, pattern avoidance, set partitions, and noncrossing partitions. We end with a section with conjectures and outlining future directions.
Kamiya, Takemura, and Terao introduced a characteristic quasi-polynomial which enumerates the numbers of elements in the complement of hyperplane arrangements modulo positive integers. In this paper, we compute the characteristic quasi-polynomials for specific arrangements which contain the Coxeter arrangements of types A, B, C, and D described by the orthonormal basis. We also compute the characteristic quasi-polynomials for their deletion arrangements and we can show that they are factorized.From this result, the poset generated by hypertori of the corresponding toric arrangement is an inductive poset.
Building sets were introduced in the study of wonderful compactifications of hyperplane arrangement complements and were later generalized to finite meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust combinatorial abstraction of convexity. Supersolvable convex geometries and antimatroids appear in the study of poset closure operators, Coxeter groups, and matroid activities. We prove that the building sets on a finite meet-semilattice form a supersolvable convex geometry. As an application, we demonstrate that building sets and nested set complexes respect certain restrictions of finite meet-semilattices unifying and extending results of several authors.
