• Order (IF 0.424) Pub Date : 2020-06-02
Csaba Biró, Bartłomiej Bosek, Heather C. Smith, William T. Trotter, Ruidong Wang, Stephen J. Young

Planar posets can have arbitrarily large dimension. However, a planar poset of height h has dimension at most 192h + 96, while a planar poset with t minimal elements has dimension at most 2t + 1. In particular, a planar poset with a unique minimal element has dimension at most 3. In this paper, we extend this result by showing that a planar poset has dimension at most 6 if it has a plane diagram in

更新日期：2020-06-02
• Order (IF 0.424) Pub Date : 2020-05-11
Noah Kravitz, Ashwin Sah

We address the following natural but hitherto unstudied question: what are the possible linear extension numbers of an n-element poset? Let LE(n) denote the set of all positive integers that arise as the number of linear extensions of some n-element poset. We show that LE(n) skews towards the “small” end of the interval [1, n!]. More specifically, LE(n) contains all of the positive integers up to $$\exp 更新日期：2020-05-11 • Order (IF 0.424) Pub Date : 2020-05-06 Mahir Bilen Can, Tien Le We continue our study of the inclusion posets of diagonal SL(n)-orbit closures in a product of two partial flag varieties. We prove that, if the diagonal action is of complexity one, then the poset is isomorphic to one of the 28 posets that we determine explicitly. Furthermore, our computations show that the number of diagonal SL(n)-orbits in any of these posets is at most 10 for any positive integer 更新日期：2020-05-06 • Order (IF 0.424) Pub Date : 2020-04-28 J. Pascal Gollin, Jakob Kneip Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order 更新日期：2020-04-28 • Order (IF 0.424) Pub Date : 2020-04-27 Martin Charles Golumbic, Vincent Limouzy We consider questions regarding the containment graphs of paths in a tree (CPT graphs), a subclass of comparability graphs, and the containment posets of paths in a tree (CPT orders). In 1984, Corneil and Golumbic observed that a graph G may be CPT, yet not every transitive orientation of G necessarily has a CPT representation, illustrating this on the even wheels W2k(k ≥ 3). Motivated by this example 更新日期：2020-04-27 • Order (IF 0.424) Pub Date : 2020-04-20 Jacob Menix, Tom Richmond If \(f:X \rightarrow X$$ is a function, the associated functional Alexandroff topology on X is the topology Pf whose closed sets are $$\{A \subseteq X : f(A) \subseteq A\}$$. We present a characterization of functional Alexandroff topologies on a finite set X and show that the collection FA(X) of all functional Alexandroff topologies on a finite set X, ordered by inclusion, is a complemented lattice

更新日期：2020-04-20
• Order (IF 0.424) Pub Date : 2020-03-26
Atrayee Majumder, Rogers Mathew, Deepak Rajendraprasad

A collection of linear orders on X, say $${\mathscr{L}}$$, is said to realize a partially ordered set (or poset) $$\mathcal {P} = (X, \preceq )$$ if, for any two distinct x,y ∈ X, x ≼ y if and only if x ≺Ly, $$\forall L \in {\mathscr{L}}$$. We call $${\mathscr{L}}$$ a realizer of $$\mathcal {P}$$. The dimension of $$\mathcal {P}$$, denoted by $$dim(\mathcal {P})$$, is the minimum cardinality of a realizer

更新日期：2020-03-26
• Order (IF 0.424) Pub Date : 2020-03-03
Jianning Su

Matoušek introduced and axiomatized a notion of symmetric difference for orthocomplemented lattices (OMLs) called the orthocomplemented difference lattics (ODLs). We focus on the class of all ODLs that are set-representable and prove that this class is not locally finite by constructing an infinite three-generated set-representable ODL. This result answers a question posed by Matoušek and Pták. We

更新日期：2020-03-03
• Order (IF 0.424) Pub Date : 2020-02-13
Giovanni Gaiffi, Viola Siconolfi

For any triple given by a positive integer n, a finite group G, and a faithful representation V of G, one can describe a subspace arrangement whose intersection lattice is a generalized Dowling lattice in the sense of Hanlon (Trans. Amer. Math. Soc. 325(1), 1–37, 1991). In this paper we construct the minimal De Concini-Procesi wonderful model associated to this subspace arrangement and give a description

更新日期：2020-02-13
• Order (IF 0.424) Pub Date : 2020-02-13
Jean-Philippe Labbé, Carsten E. M. C. Lange

We study the size of certain acyclic domains that arise from geometric and combinatorial constructions. These acyclic domains consist of all permutations visited by commuting equivalence classes of maximal reduced decompositions if we consider the symmetric group and, more generally, of all c-singletons of a Cambrian lattice associated to the weak order of a finite Coxeter group. For this reason, we

更新日期：2020-02-13
• Order (IF 0.424) Pub Date : 2020-02-10
Dániel Gerbner, Balázs Keszegh, Balázs Patkós

A subfamily $$\{F_{1},F_{2},\dots ,F_{|P|}\}\subseteq \mathcal {F}$$ of sets is a copy of a poset P in $$\mathcal {F}$$ if there exists a bijection $$\phi :P\rightarrow \{F_{1},F_{2},\dots ,F_{|P|}\}$$ such that whenever $$x \le _{P} x^{\prime }$$ holds, then so does $$\phi (x)\subseteq \phi (x^{\prime })$$. For a family $$\mathcal {F}$$ of sets, let $$c(P,\mathcal {F})$$ denote the number of copies

更新日期：2020-02-10
• Order (IF 0.424) Pub Date : 2020-01-22
Claudia Mureşan, Júlia Kulin

We investigate the possible values of the numbers of congruences of finite lattices of an arbitrary but fixed cardinality. Motivated by a result of Freese and continuing Czédli’s recent work, we prove that the third, fourth and fifth largest numbers of congruences of an n–element lattice are: 5 ⋅ 2n− 5 if n ≥ 5, 2n− 3 and 7 ⋅ 2n− 6 if n ≥ 6, respectively. We also determine the structures of the n–element

更新日期：2020-01-22
• Order (IF 0.424) Pub Date : 2020-01-10
M. E. Adams, H. P. Sankappanavar, Júlia Vaz de Carvalho

In this paper, we investigate the varieties Mn and Kn of regular pseudocomplemented de Morgan and Kleene algebras of range n, respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in Mn and explicitly describe the dual spaces of the simple algebras in M1 and K1. We

更新日期：2020-01-10
• Order (IF 0.424) Pub Date : 2019-12-27
Masato Kobayashi

We generalize the author’s formula (2011) on weighted counting of inversions on permutations to one on alternating sign matrices. The proof is based on the sequential construction of alternating sign matrices from the unit matrix which essentially follows from the earlier work of Lascoux-Schützenberger (1996).

更新日期：2019-12-27
• Order (IF 0.424) Pub Date : 2019-12-18
Gábor Czédli

Let L be a finite n-element semilattice. We prove that if L has at least 127 ⋅ 2n− 8 subsemilattices, then L is planar. For n > 8, this result is sharp since there is a non-planar semilattice with exactly 127 ⋅ 2n− 8 − 1 subsemilattices.

更新日期：2019-12-18
• Order (IF 0.424) Pub Date : 2019-12-13
Themba Dube

We say a prime element of an algebraic frame is amenable if it comparable to every compact element. If every prime element of an algebraic frame L is amenable, we say L is an amenable frame. If the localization of L at every prime element is amenable, we say L is locally amenable. These concepts are motivated by notions of divided and locally divided commutative rings. We show that an algebraic frame

更新日期：2019-12-13
• Order (IF 0.424) Pub Date : 2019-12-09
Reinhard Diestel, Jakob Kneip

Separation systems are posets with additional structure that form an abstract setting in which tangle-like clusters in graphs, matroids and other combinatorial structures can be expressed and studied. This paper offers some basic theory about infinite separation systems and how they relate to the finite separation systems they induce. They can be used to prove tangle-type duality theorems for infinite

更新日期：2019-12-09
• Order (IF 0.424) Pub Date : 2019-11-22
Joshua Cooper, Peter Gartland, Hays Whitlatch

Hasebe and Tsujie characterized the set of induced N-free and bowtie-free posets as a certain class of recursively defined subposets which they term “$$\mathcal {V}$$-posets”. Here we offer a new characterization of $$\mathcal {V}$$-posets by introducing a property we refer to as autonomy. A poset $$\mathcal {P}$$ is said to be autonomous if there exists a directed acyclic graph D (with adjacency matrix

更新日期：2019-11-22
• Order (IF 0.424) Pub Date : 2019-11-19
Nathan Bowler, Jakob Kneip

Abstract separation systems are a new unifying framework in which separations of graph, matroids and other combinatorial structures can be expressed and studied. We characterize the abstract separation systems that have representations as separation systems of graphs, sets, or set bipartitions.

更新日期：2019-11-19
• Order (IF 0.424) Pub Date : 2019-11-19
D. Georgiou, S. Iliadis, A. Megaritis, F. Sereti

The universality property plays an important role in the field of frames and the notion of saturated class of frames is combined with this property (see Dube et al. (Topology and its Applications 160, 2454–2464, 2013); Iliadis (Topology and its Applications 179, 99–110, 2015) and Iliadis (Topology and its Applications 201, 92–109, 18)). In this paper, we continue such a study, introducing and studying

更新日期：2019-11-19
• Order (IF 0.424) Pub Date : 2019-11-16

For a finite subset M ⊂ [x1,…, xd] of monomials, we describe how to constructively obtain a monomial ideal $$I\subseteq R = K[x_{1},\ldots ,x_{d}]$$ such that the set of monomials in Soc(I) ∖ I is precisely M, or such that $$\overline {M}\subseteq R/I$$ is a K-basis for the the socle of R/I. For a given M we obtain a natural class of monomials ideals I with this property. This is done by using solely

更新日期：2019-11-16
• Order (IF 0.424) Pub Date : 2019-07-20
Gergő Gyenizse

In this paper, we investigate the class of lattices representable with posets satisfying the DCC condition. We describe a way to decide whether a finite lattice is in this class. We also give a necessary condition for an arbitrary lattice to be in this class. This hints at a notion that would be a weaker version of lower boundedness.

更新日期：2019-07-20
• Order (IF 0.424) Pub Date : 2019-06-26
Miloš S. Kurilić, Stevo Todorčević

We show that the poset of copies $$\mathbb {P} (\mathbb {Q}_{n} )=\langle \{ f[X]: f\in \text {Emb} (\mathbb {Q}_{n} ) \},\subset \rangle$$ of the countable homogeneous universal n-labeled linear order, $$\mathbb {Q}_{n}$$, is forcing equivalent to the poset $$\mathbb {S} \ast \pi$$, where $$\mathbb {S}$$ is the Sacks perfect set forcing and $$1_{\mathbb {S}} \Vdash  \pi$$ is an atomless separative

更新日期：2019-06-26
• Order (IF 0.424) Pub Date : 2019-06-10
Bernd S. W. Schröder

We prove that, for a finite ordered set P that contains no crowns with 6 or more elements, it can be determined in polynomial time if P has the fixed point property. This result is obtained by proving that every such ordered set must contain a point of rank 1 that has a unique lower cover or a retractable minimal element.

更新日期：2019-06-10
• Order (IF 0.424) Pub Date : 2019-06-10
Johannes Marti, Riccardo Pinosio

In this paper we present a duality between nonmonotonic consequence relations and well-founded convex geometries. On one side of the duality we consider nonmonotonic consequence relations satisfying the axioms of an infinitary variant of System P, which is one of the most studied axiomatic systems for nonmonotonic reasoning, conditional logic and belief revision. On the other side of the duality we

更新日期：2019-06-10
• Order (IF 0.424) Pub Date : 2019-06-10
Henri Mühle, Vivien Ripoll

We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of an element in a generated group. While one form of connectivity essentially reflects the connectivity of the poset diagram, the other two are a bit more involved: Hurwitz-connectivity has its origins in algebraic geometry, and shellability in topology. We propose a framework

更新日期：2019-06-10
• Order (IF 0.424) Pub Date : 2019-06-07

A poset $$\mathbb {P}$$ is called reversible iff every bijective homomorphism $$f:\mathbb {P} \rightarrow \mathbb {P}$$ is an automorphism. Let $$\mathcal {W}$$ and $$\mathcal {W}^{*}$$ denote the classes of well orders and their inverses respectively. We characterize reversibility in the class of posets of the form $$\mathbb {P} =\bigcup _{i\in I}\mathbb {L}_{i}$$, where $$\mathbb {L}_{i}, i\in I$$

更新日期：2019-06-07
• Order (IF 0.424) Pub Date : 2019-05-25
Wenfeng Zhang, Xiaoquan Xu

Given any subset selection $$\mathcal {Z}$$ for posets, we study two weakenings of the known concept of $$\mathcal {Z}$$-predistributivity, namely, $$\mathcal {Z}$$-quasidistributivity and $$\mathcal {Z}$$-meet-distributivity. The former generalizes quasicontinuity, and the latter meet-continuity of complete lattices. We show for global completions $$\mathcal {Z}$$ that the $$\mathcal {Z}$$-quasidistributive

更新日期：2019-05-25
• Order (IF 0.424) Pub Date : 2019-05-23
Noga Alon, Omri Ben-Eliezer

It was recently proved in Alon et al. (2017) that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following ordered matrix removal lemma: For any finite alphabet Γ, any hereditary property $$\mathcal {P}$$ of matrices over Γ, and any 𝜖 > 0, there exists \(f_{\mathcal

更新日期：2019-05-23
• Order (IF 0.424) Pub Date : 2019-05-04
Jimmy Devillet, Bruno Teheux

We characterize the associative, idempotent, symmetric, and order-preserving binary operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In particular, we prove that the number of associative, idempotent, symmetric, and order-preserving operations on an n-element chain is the nth Catalan number.

更新日期：2019-05-04
• Order (IF 0.424) Pub Date : 2019-05-01
Ivan Chajda, Helmut Länger, Jan Paseka

In Chajda and Länger (Math. Bohem. 143, 89–97, 2018) the concept of relative pseudocomplementation was extended to posets. We introduce the concept of a congruence in a relatively pseudocomplemented poset within the framework of Hilbert algebras and we study under which conditions the quotient structure is a relatively pseudocomplemented poset again. This problem is solved e.g. for finite or linearly

更新日期：2019-05-01
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