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Constructions of Codes with Weighted Poset Block Metrics Order (IF 0.4) Pub Date : 2024-03-14 Wen Ma, Jinquan Luo
Weighted poset block metric is a generalization of two types of metrics: one is weighted poset metric introduced by Panek and Pinheiro (2010) and the other is metric for linear error-block codes introduced by Feng and Hickernell (2006). This type of metrics includes many classical metrics such as Hamming metric, Lee metric, poset metric, pomset metric, poset block metric, pomset block metric and so
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Boolean Dimension of a Boolean Lattice Order (IF 0.4) Pub Date : 2024-03-14 Marcin Briański, Jȩdrzej Hodor, Hoang La, Piotr Micek, Katzper Michno
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On a Problem of Conrad on Riesz Space Structures Order (IF 0.4) Pub Date : 2024-03-06 Giacomo Lenzi
This paper is concerned with Riesz space structures on a lattice ordered abelian group, continuing a line of research conducted by the author and the collaborators Antonio Di Nola and Gaetano Vitale. First we prove a statement in a paper of Paul Conrad (given without proof) that every non-archimedean totally ordered abelian group has at least two Riesz space structures, if any. Then, as a main result
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Poset Ramsey Number $$R(P,Q_n)$$ . II. N-Shaped Poset Order (IF 0.4) Pub Date : 2024-02-28 Maria Axenovich, Christian Winter
Given partially ordered sets (posets) \((P, \le _P)\) and \((P', \le _{P'})\), we say that \(P'\) contains a copy of P if for some injective function \(f:P\rightarrow P'\) and for any \(A, B\in P\), \(A\le _P B\) if and only if \(f(A)\le _{P'} f(B)\). For any posets P and Q, the poset Ramsey number R(P, Q) is the least positive integer N such that no matter how the elements of an N-dimensional Boolean
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Decidability of Well Quasi-Order and Atomicity for Equivalence Relations Under Embedding Orderings Order (IF 0.4) Pub Date : 2024-02-14
Abstract We consider the posets of equivalence relations on finite sets under the standard embedding ordering and under the consecutive embedding ordering. In the latter case, the relations are also assumed to have an underlying linear order, which governs consecutive embeddings. For each poset we ask the well quasi-order and atomicity decidability questions: Given finitely many equivalence relations
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Higman’s Lemma is Stronger for Better Quasi Orders Order (IF 0.4) Pub Date : 2024-01-23 Anton Freund
We prove that Higman’s lemma is strictly stronger for better quasi orders than for well quasi orders, within the framework of reverse mathematics. In fact, we show a stronger result: the infinite Ramsey theorem (for tuples of all lengths) is equivalent to the statement that any array \([\mathbb N]^{n+1}\rightarrow \mathbb N^n\times X\) for a well order X and \(n\in \mathbb N\) is good, over the base
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Jordan-Hölder Theorem with Uniqueness for Semimodular Lattices Order (IF 0.4) Pub Date : 2024-01-11 Pavel Paták
In 2011 Czédli and Schmidt proved the strongest form of Jordan-Hölder theorem for lattices, which they called Jordan-Hölder theorem with uniqueness: Given two maximal chains in a semimodular lattice of finite height, they both have the same length and there is a unique bijection that takes the prime intervals of the first chain to the prime intervals of the second chain such that the interval and its
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On Dually-CPT and Strongly-CPT Posets Order (IF 0.4) Pub Date : 2023-12-20 Liliana Alcón, Martin Charles Golumbic, Noemí Gudiño, Marisa Gutierrez, Vincent Limouzy
A poset is a containment of paths in a tree (CPT) if it admits a representation by containment where each element of the poset is represented by a path in a tree and two elements are comparable in the poset if the corresponding paths are related by the inclusion relation. Recently Alcón, Gudiño and Gutierrez (Discrete Applied Math. 245, 139–147, 2018) introduced proper subclasses of CPT posets, namely
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A Study on Some Classes of Distributive Lattices with a Generalized Implication Order (IF 0.4) Pub Date : 2023-11-29 Ismael Calomino, Jorge Castro, Sergio Celani, Luciana Valenzuela
A generalized implication on a distributive lattice \(\varvec{A}\) is a function between \(\varvec{A} \times \varvec{A}\) to ideals of \(\varvec{A}\) satisfying similar conditions to strict implication of weak Heyting algebras. Relative anihilators and quasi-modal operators are examples of generalized implication in distributive lattices. The aim of this paper is to study some classes of distributive
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Antichains in the Bruhat Order for the Classes $$\mathcal {A}(n,k)$$ Order (IF 0.4) Pub Date : 2023-11-28 Henrique F. da Cruz
Let \(\varvec{\mathcal {A}(n,k)}\) represent the collection of all \(\varvec{n\times n}\) zero-and-one matrices, with the sum of all rows and columns equalling \(\varvec{k}\). This set can be ordered by an extension of the classical Bruhat order for permutations, seen as permutation matrices. The Bruhat order on \(\varvec{\mathcal {A}(n,k)}\) differs from the Bruhat order on permutations matrices not
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On the Rigidity of Lattices of Topologies on Vector Spaces Order (IF 0.4) Pub Date : 2023-11-22 Takanobu Aoyama
A vector topology on a vector space over a topological field is a (not necessarily Hausdorff) topology by which the addition and the scalar multiplication are continuous. We prove that, if an isomorphism between the lattices of topologies of two vector spaces preserves vector topologies, then the isomorphism is induced by a translation, a semilinear isomorphism and the complement map. As a consequence
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A Proof of the Alternate Thomassé Conjecture for Countable N-Free Posets Order (IF 0.4) Pub Date : 2023-11-20 Davoud Abdi
An N-free poset is a poset whose comparability graph does not embed an induced path with four vertices. We use the well-quasi-order property of the class of countable N-free posets and some labelled ordered trees to show that a countable N-free poset has one or infinitely many siblings, up to isomorphism. This, partially proves a conjecture stated by Thomassé for this class.
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The Dimension of Divisibility Orders and Multiset Posets Order (IF 0.4) Pub Date : 2023-11-22 Milan Haiman
The Dushnik–Miller dimension of a poset P is the least d for which P can be embedded into a product of d chains. Lewis and Souza isibility order on the interval of integers \([N/\kappa , N]\) is bounded above by \(\kappa (\log \kappa )^{1+o(1)}\) and below by \(\Omega ((\log \kappa /\log \log \kappa )^2)\). We improve the upper bound to \(O((\log \kappa )^3/(\log \log \kappa )^2).\) We deduce this
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$$\mathfrak {m}$$ -Baer and $$\mathfrak {m}$$ -Rickart Lattices Order (IF 0.4) Pub Date : 2023-11-06 Mauricio Medina-Bárcenas, Hugo Rincón Mejía
In this paper we introduce the notions of Rickart and Baer lattices and their duals. We show that part of the theory of Rickart and Baer modules can be understood just using techniques from the theory of lattices. For, we use linear morphisms introduced by T. Albu and M. Iosif. We focus on a submonoid with zero \(\mathfrak {m}\) of the monoid of all linear endomorphism of a lattice \(\mathcal {L}\)
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Representations of Finite Groups by Posets of Small Height Order (IF 0.4) Pub Date : 2023-10-28 Gergő Gyenizse, Péter Hajnal, László Zádori
We characterize the finite groups as the automorphism groups of the finite height one posets with at most four orbits. We also prove that for each \(n\ge 8\), the cyclic group \(\textbf{Z}_n\) is isomorphic to the automorphism group of a finite height one poset with at most two orbits. As a consequence, for each n, we determine the minimum size of the posets whose automorphism groups are isomorphic
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General Mixed Lattices Order (IF 0.4) Pub Date : 2023-09-27 Jani Jokela
A mixed lattice is a lattice-type structure consisting of a set with two partial orderings, and generalizing the notion of a lattice. Mixed lattice theory has previously been studied in various algebraic structures, such as groups and semigroups, while the more general notion of a mixed lattice remains unexplored. In this paper, we study the fundamental properties of mixed lattices and the relationships
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A short note on the characterization of countable chains with finite big Ramsey spectra Order (IF 0.4) Pub Date : 2023-09-11 Keegan Dasilva Barbosa, Dragan Mašulović, Rajko Nenadov
In this short note we confirm the deep structural correspondence between the complexity of a countable scattered chain (\(=\) strict linear order) and its big Ramsey combinatorics: we show that a countable scattered chain has finite big Ramsey degrees if and only if it is of finite Hausdorff rank. This also provides a complete characterization of countable chains whose big Ramsey spectra are finite
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Interleaving by Parts: Join Decompositions of Interleavings and Join-Assemblage of Geodesics Order (IF 0.4) Pub Date : 2023-09-08 Woojin Kim, Facundo Mémoli, Anastasios Stefanou
Metrics of interest in topological data analysis (TDA) are often explicitly or implicitly in the form of an interleaving distance \({d_{\textrm{I}}}\) between poset maps (i.e. order-preserving maps), e.g. the Gromov-Hausdorff distance between metric spaces can be reformulated in this way. We propose a representation of a poset map \(\textbf{F}:{\mathcal {P}}\rightarrow {\mathcal {Q}}\) as a join (i
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Complementary Choice Functions Order (IF 0.4) Pub Date : 2023-08-31 Danilov V. I.
The paper studies complementary choice functions, i.e. monotonic and consistent choice functions. Such choice functions were introduced and used in the work [13] for investigation of matchings with complementary contracts. We provide three (universal) ways of constructing such choice functions: through pre-topologies, as direct images of completely complementary (or pre-ordered) choice functions, and
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Maximal Cliques Lattices Structures for Cocomparability Graphs with Algorithmic Applications Order (IF 0.4) Pub Date : 2023-08-08 Jérémie Dusart, Michel Habib, Derek G. Corneil
A cocomparability graph is a graph whose complement admits a transitive orientation. An interval graph is the intersection graph of a family of intervals on the real line. In this paper we investigate the relationships between interval and cocomparability graphs. This study is motivated by recent results (Corneil et al., SIAM J. Comput. 42(3):792–807, 2013; Dusart et al., SIAM J. Comput. 50(3):1148–1153
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Chain-dependent Conditions in Extremal Set Theory Order (IF 0.4) Pub Date : 2023-08-07 Dániel T. Nagy, Kartal Nagy
In extremal set theory our usual goal is to find the maximal size of a family of subsets of an n-element set satisfying a condition. A condition is called chain-dependent, if it is satisfied for a family if and only if it is satisfied for its intersections with the n! full chains. We introduce a method to handle problems with such conditions, then show how it can be used to prove three classic theorems
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Injective Hulls are Completions of Ordered Algebras Order (IF 0.4) Pub Date : 2023-08-03 Xia Zhang, Valdis Laan, Jianjun Feng
It is well-known that the Dedekind-MacNeille completion of a poset is its injective hull. We prove that the injective hull of an ordered universal algebra with respect to a specific class of monomorphisms has properties that are similar to the properties of the Dedekind-MacNeille completion of a poset. In particular, this injective hull induces a reflector functor from the category of ordered algebras
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Filter Classes of Upsets of Distributive Lattices Order (IF 0.4) Pub Date : 2023-08-02 Adam Přenosil
Let us say that a class of upward closed sets (upsets) of distributive lattices is a finitary filter class if it is closed under homomorphic preimages, intersections, and directed unions. We show that the only finitary filter classes of upsets of distributive lattices are formed by what we call n-filters. These are related to the finite Boolean lattice with n atoms in the same way that filters are
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On the Automorphism Group of the Substructure Ordering of Finite Directed Graphs Order (IF 0.4) Pub Date : 2023-07-25 Fanni K. Nedényi, Ádám Kunos
We investigate the automorphism group of the substructure ordering of finite directed graphs. The second author conjectured that it is isomorphic to the 768-element group \((\mathbb {Z}_2^4 \times S_4)\rtimes _{\alpha } \mathbb {Z}_2\). Though unable to prove it, we solidify this conjecture by showing that the automorphism group behaves as expected by the conjecture on the first few levels of the poset
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Extension of Sectional Pseudocomplementation in Posets Order (IF 0.4) Pub Date : 2023-07-22 Jānis Cīrulis
Sectional pseudocomplementation (sp-complementation) on a poset is a partial operation \(*\) which associates with every pair (x, y) of elements, where \(x \ge y\), the pseudocomplement \(x*y\) of x in the upper section [y). Any total extension \(\rightarrow \) of \(*\) is said to be an extended sp-complementation and is considered as an implication-like operation. Extended sp-complementations have
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Shattering k-sets with Permutations Order (IF 0.4) Pub Date : 2023-06-20 J. Robert Johnson, Belinda Wickes
Many concepts from extremal set theory have analogues for families of permutations. This paper is concerned with the notion of shattering for permutations. A family \(\mathcal {P}\) of permutations of an n-element set X shatters a k-set from X if it appears in each of the k! possible orders in some permutation in \(\mathcal {P}\). The smallest family \(\mathcal {P}\) which shatters every k-subset of
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A Study of the Small Inductive Dimension in the Area of Finite Lattices Order (IF 0.4) Pub Date : 2023-06-17 D. Georgiou, A. Megaritis, G. Prinos, F. Sereti
The Dimension Theory in the class of frames was the base of many researches. Especially, the covering dimension, the quasi covering dimension, the small inductive dimension and the large inductive dimension for frames have been studied extensively (see (Banaschewski and Gilmour J. London Math. Soc. 39(2) 1–8 1989), (Brijlall and Baboolal Indian J. Pure Appl. Math. 39(5), 375–402 2008), (Brijlall and
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The Dedekind-MacNeille Completion of a Poset as a Set of Ideals of the Incidence Algebra Order (IF 0.4) Pub Date : 2023-06-14 Manfred Dugas, Daniel Herden
We show that the Dedekind-MacNeille completion of any poset \((P,\le )\) can be recovered by algebraic means from the ideal structure of the finitary incidence algebra FI(P) over a commutative unital indecomposable ring R.
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Two Flags in a Semimodular Lattice Generate an Antimatroid Order (IF 0.4) Pub Date : 2023-06-13 Koyo Hayashi, Hiroshi Hirai
A basic property in a modular lattice is that any two flags generate a distributive sublattice. It is shown (Abels 1991, Herscovici 1998) that two flags in a semimodular lattice no longer generate such a good sublattice, whereas shortest galleries connecting them form a relatively good join-sublattice. In this note, we sharpen this investigation to establish an analogue of the two-flag generation theorem
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Poset Ramsey Number $$R(P,Q_{n})$$ . I. Complete Multipartite Posets Order (IF 0.4) Pub Date : 2023-05-31 Christian Winter
A poset \((P^{\prime },\le _{P^{\prime }})\) contains a copy of some other poset \((P,\le _{P})\) if there is an injection \(f:P'\rightarrow P\) where for every \(X,Y\in P\), \(X\le _{P} Y\) if and only if \(f(X)\le _{P'} f(Y)\). For any posets P and Q, the poset Ramsey number R(P, Q) is the smallest integer N such that any blue/red coloring of a Boolean lattice of dimension N contains either a copy
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Weakenings of the Knaster property Order (IF 0.4) Pub Date : 2023-06-01 Júnio Luan Pereira
Our intent in this work is to propose statements for subclasses of productively ccc partial orders that act as weak versions of the well-known Knaster property. It introduces an intuitive reasoning for the statements chosen and makes use of works from Kunen and Todorčević to present two intrinsically different productively ccc partial orders satisfying such weak statements, but not the Knaster one
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Gluing Residuated Lattices Order (IF 0.4) Pub Date : 2023-05-31 Nikolaos Galatos, Sara Ugolini
We introduce and characterize various gluing constructions for residuated lattices that intersect on a common subreduct, and which are subalgebras, or appropriate subreducts, of the resulting structure. Starting from the 1-sum construction (also known as ordinal sum for residuated structures), where algebras that intersect only in the top element are glued together, we first consider the gluing on
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Algebras, Graphs and Ordered Sets - ALGOS 2020 & the Mathematical Contributions of Maurice Pouzet. Order (IF 0.4) Pub Date : 2023-05-17 Miguel Couceiro,Dwight Duffus
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Helly-Type Theorems for the Ordering of the Vertices of a Hypergraph Order (IF 0.4) Pub Date : 2023-05-16 Csaba Biró, Jenő Lehel, Géza Tóth
Let H be a complete r-uniform hypergraph such that two vertices are marked in each edge as its ‘boundary’ vertices. A linear ordering of the vertex set of H is called an agreeing linear order, provided all vertices of each edge of H lie between its two boundary vertices. We prove the following Helly-type theorem: if there is an agreeing linear order on the vertex set of every subhypergraph of H with
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Separability, Boxicity, and Partial Orders Order (IF 0.4) Pub Date : 2023-05-15 José-Miguel Díaz-Báñez, Paul Horn, Mario A. Lopez, Nestaly Marín, Adriana Ramírez-Vigueras, Oriol Solé-Pi, Alex Stevens, Jorge Urrutia
A collection \(S=\{S_i, \ldots , S_n\}\) of disjoint closed convex sets in \(\mathbb {R}^d\) is separable if there exists a direction (a non-zero vector) \( \overrightarrow{v}\) of \(\mathbb {R}^d\) such that the elements of S can be removed, one at a time, by translating them an arbitrarily large distance in the direction \( \overrightarrow{v}\) without hitting another element of S. We say that \(S_i
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Characteristic Polynomials of the Weak Order on Classical and Affine Coxeter Groups Order (IF 0.4) Pub Date : 2023-05-13 Jang Soo Kim, Sun-mi Yun
We find a simple product formula for the characteristic polynomial of the permutations with a fixed descent set under the weak order. As a corollary we obtain a simple product formula for the characteristic polynomial of alternating permutations. We generalize these results to Coxeter groups. We also find a formula for the generating function for the characteristic polynomials of classical Coxeter
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An Approximation Algorithm for Random Generation of Capacities Order (IF 0.4) Pub Date : 2023-05-02 Michel Grabisch, Christophe Labreuche, Peiqi Sun
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The Coincidence of the Bruhat Order and the Secondary Bruhat Order on $$\mathcal {A}(n,k)$$ Order (IF 0.4) Pub Date : 2023-04-27 Tao Zhang, Houyi Yu
Given a positive integer n and a nonnegative integer k with \(k\le n\), we denote by \(\mathcal {A}(n,k)\) the class of all n-by-n (0, 1)-matrices with constant row and column sums k. In this paper, we show that the Bruhat order and the secondary Bruhat order coincide on \(\mathcal {A}(n,k)\) if and only if either \(0\le n\le 5\) or \(k\in \{0,1,2,n-2,n-1,n\}\) with \(n\ge 6\).
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Improved Lower Bounds on the On-line Chain Partitioning of Posets of Bounded Dimension Order (IF 0.4) Pub Date : 2023-03-29 Csaba Biró, Israel R Curbelo
An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemerédi proved that any on-line algorithm could be forced to use \(\left( {\begin{array}{c}w+1\\ 2\end{array}}\right)\) chains to partition a poset of width w. The maximum number of chains that can be forced on any on-line algorithm remains
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F3-Reconstruction and Bi-Founded 2-Structures Order (IF 0.4) Pub Date : 2023-02-27 Youssef Boudabbous, Christian Delhommé
Two binary relational structures of a same signature are (≤ k)-hypomorphic if they have the same vertex set and their restrictions to each set of at most k vertices are isomorphic. A binary relational structure is (≤ k)-reconstructible if it is isomorphic with each structure it is (≤ k)-hypomorphic with. We establish an inductive characterisation of pairs of (≤ 3)-hypomorphic binary relational structures
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Geometric Realizations of Tamari Interval Lattices Via Cubic Coordinates Order (IF 0.4) Pub Date : 2023-02-23 Camille Combe
We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with Tamari intervals. We show that in each degree the set of cubic coordinates forms a lattice, isomorphic to the lattice of Tamari intervals. Geometric realizations are naturally obtained by placing cubic coordinates
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Order-Preserving Self-Maps of Complete Lattices Order (IF 0.4) Pub Date : 2023-02-13 Bernhard Ganter
We study isotone self-maps of complete lattices and their fixed point sets, which are complete lattices contained as suborders, but not necessarily as subsemilattices. We develop a representation of such maps by means of relations and show how to navigate their fixed point lattices using a modification of the standard Next closure algorithm. Our approach is inspired by early work of Shmuely (J. Combin
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The Saturation Spectrum for Antichains of Subsets Order (IF 0.4) Pub Date : 2023-02-10 Jerrold R. Griggs, Thomas Kalinowski, Uwe Leck, Ian T. Roberts, Michael Schmitz
Extending a classical theorem of Sperner, we characterize the integers m such that there exists a maximal antichain of size m in the Boolean lattice Bn, that is, the power set of \([n]:=\{1,2,\dots ,n\}\), ordered by inclusion. As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t
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Stability of Boolean Function Classes with Respect to Clones of Linear Functions Order (IF 0.4) Pub Date : 2023-02-09 Miguel Couceiro, Erkko Lehtonen
We consider classes of Boolean functions stable under compositions both from the right and from the left with clones. Motivated by the question how many properties of Boolean functions can be defined by means of linear equations, we focus on stability under compositions with the clone of linear idempotent functions. It follows from a result by Sparks that there are countably many such linearly definable
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Notes on Sharp and Principal Elements in Effect Algebras Order (IF 0.4) Pub Date : 2023-01-17 Yali Wu, Xia Li, Jing Wang
In this paper, we characterize the sharp elements and principal elements in effect algebras. Furthermore, the sufficient and necessary conditions for ES (the set of sharp elements in an effect algebra E) to be an orthoalgebra, and to be an orthomodular poset are given. We also answer the open problems raised in 1996 by Gudder. Moreover, we show that in effect algebras, every pre-lattice ideal is an
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Combinatorics Arising from Lax Colimits of Posets Order (IF 0.4) Pub Date : 2023-01-04 Zurab Janelidze, Helmut Prodinger, Francois van Niekerk
In this paper we study maximal chains in certain lattices constructed from powers of chains by iterated lax colimits in the 2-category of posets. Such a study is motivated by the fact that in lower dimensions, we get some familiar combinatorial objects such as Dyck paths and Kreweras walks.
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Orthogonality Spaces Associated with Posets Order (IF 0.4) Pub Date : 2022-12-23 Gejza Jenča
An orthogonality space is a set equipped with a symmetric, irreflexive relation called orthogonality. Every orthogonality space has an associated complete ortholattice, called the logic of the orthogonality space. To every poset, we associate an orthogonality space consisting of proper quotients (that means, nonsingleton closed intervals), equipped with a certain orthogonality relation. We prove that
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Sectionable Tournaments: their Topology and Coloring Order (IF 0.4) Pub Date : 2022-12-07 Zakir Deniz
We provide a detailed study of topological and combinatorial properties of sectionable tournaments. This class forms an inductively constructed family of tournaments grounded over simply disconnected tournaments, those tournaments whose fundamental groups of acyclic complexes are non-trivial. When T is a sectionable tournament, we fully describe the cell-structure of its acyclic complex Acy(T) by using
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Length-preserving Extensions of a Semimodular Lattice by Lowering a Join-irreducible Element Order (IF 0.4) Pub Date : 2022-11-28 Gábor Czédli
We extend the bijective correspondence between finite semimodular lattices and Faigle geometries to an analogous correspondence between semimodular lattices of finite lengths and a larger class of geometries. As the main application, we prove that if e is a join-irreducible element of a semimodular lattice L of finite length and h < e in L such that e does not cover h, then e can be “lowered” to a
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On Nontrivial Weak Dicomplementations and the Lattice Congruences that Preserve Them Order (IF 0.4) Pub Date : 2022-11-29 Leonard Kwuida, Claudia Mureşan
We study the existence of nontrivial and of representable (dual) weak complementations, along with the lattice congruences that preserve them, in different constructions of bounded lattices, then use this study to determine the finite (dual) weakly complemented lattices with the largest numbers of congruences, along with the structures of their congruence lattices. It turns out that, if \(n\ge 7\)
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A Symmetric-Difference-Closed Orthomodular Lattice That Is Stateless Order (IF 0.4) Pub Date : 2022-11-24 Václav Voráček, Pavel Pták
This paper carries on the investigation of the orthomodular lattices that are endowed with a symmetric difference. Let us call them ODLs. Note that the ODLs may have a certain bearing on “quantum logics” - the ODLs are close to Boolean algebras though they capture the phenomenon of non-compatibility. The initial question in studying the state space of the ODLs is whether the state space can be poor
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Choice Functions on Posets Order (IF 0.4) Pub Date : 2022-11-01 Danilov V. I.
In the paper we study choice functions on posets satisfying the conditions of heredity and outcast. For every well-ordered sequence of elements of a poset, we define the corresponding ‘elementary’ choice function. Every such choice function satisfies the conditions of heredity and outcast. Inversely, every choice function satisfying the conditions of heredity and outcast can be represented as a union
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On Finitely-Generated Johansson Algebras Order (IF 0.4) Pub Date : 2022-10-18 Alex Citkin
Kuznetsov’s Theorem about finitely-generated Heyting algebras has been extended to Johansson algebras in the following way: if \(\mathbf {A} = (\mathsf {A}; \wedge ,\vee ,\rightarrow ,\mathbf {1},\mathbf {f})\) is a Johansson algebra, by the rank of element \(\mathsf {a} \in \mathsf {A}\), we understand the cardinality of the set \(\{\mathsf {b} \in \mathbf {A} | \mathsf {b} \le \mathsf {a} \}\), and
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The Canonical Complex of the Weak Order Order (IF 0.4) Pub Date : 2022-10-15 Doriann Albertin, Vincent Pilaud
We define and study the canonical complex of a finite semidistributive lattice L. It is the simplicial complex on the join or meet irreducible elements of L which encodes each interval of L by recording the canonical join representation of its bottom element and the canonical meet representation of its top element. This complex behaves properly with respect to lattice quotients of L, in the sense that
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The Number of Subuniverses, Congruences, Weak Congruences of Semilattices Defined by Trees Order (IF 0.4) Pub Date : 2022-10-14 Delbrin Ahmed, Eszter K. Horváth, Zoltán Németh
We determine the number of subuniverses of semilattices defined by arbitrary and special kinds of trees via combinatorial considerations. Using a result of Freese and Nation, we give a formula for the number of congruences of semilattices defined by arbitrary and special kinds of trees. Using both results, we prove a formula for the number of weak congruences of semilattices defined by a binary tree;
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A Construction for Boolean Cube Ramsey Numbers Order (IF 0.4) Pub Date : 2022-10-08 Tom Bohman, Fei Peng
Let Qn be the poset that consists of all subsets of a fixed n-element set, ordered by set inclusion. The poset cube Ramsey number R(Qn,Qn) is defined as the least m such that any 2-coloring of the elements of Qm admits a monochromatic copy of Qn. The trivial lower bound R(Qn,Qn) ≥ 2n was improved by Cox and Stolee, who showed R(Qn,Qn) ≥ 2n + 1 for 3 ≤ n ≤ 8 and n ≥ 13 using a probabilistic existence
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Polymatroids, Closure Operators and Lattices Order (IF 0.4) Pub Date : 2022-10-01 William Gustafson
In this article we study the closure operators of polymatroids from a lattice theoretic point of view. We show that polymatroid closure operators relate to lattices enriched with a generating set in the same way matroids relate to geometric lattices. Through this relation we define a notion of minors for lattices enriched with a generating set. For the lattice of flats of a graphic matroid, the minors
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Symmetric Maximal Condorcet Domains Order (IF 0.4) Pub Date : 2022-09-24 Alexander Karpov, Arkadii Slinko
We introduce the operation of composition of domains and show that it reduces the classification of symmetric maximal Condorcet domains to the indecomposable ones. The only non-trivial indecomposable symmetric maximal domains known are the domains consisting of four linear orders examples of which were given by Raynaud (1981) and Danilov and Koshevoy (Order 30(1), 181–194 2013). We call them Raynaud
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A Point-Free Approach to Canonical Extensions of Boolean Algebras and Bounded Archimedean $$\ell$$ ℓ -Algebras Order (IF 0.4) Pub Date : 2022-09-19 G. Bezhanishvili, L. Carai, P. Morandi
Recently W. Holliday gave a choice-free construction of a canonical extension of a boolean algebra B as the boolean algebra of regular open subsets of the Alexandroff topology on the poset of proper filters of B. We make this construction point-free by replacing the Alexandroff space of proper filters of B with the free frame \(\mathcal {L}_B\) generated by the bounded meet-semilattice of all filters
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Big Ramsey Spectra of Countable Chains Order (IF 0.4) Pub Date : 2022-08-08 Dragan Mašulović
A big Ramsey spectrum of a countable chain (i.e. strict linear order) C is a sequence of big Ramsey degrees of finite chains computed in C. In this paper we consider big Ramsey spectra of countable scattered chains. We prove that countable scattered chains of infinite Hausdorff rank do not have finite big Ramsey spectra, and that countable scattered chains of finite Hausdorff rank with bounded finite