Order Ideals Research Articles

On the height of order ideals

On the height of order ideals

Boolean complexes and boolean numbers

The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of these spheres is the boolean number, which can be computed inductively from the unlabeled Coxeter system, thus defining a graph invariant. For certain families of graphs, the boolean numbers have intriguing combinatorial properties. This work involves joint efforts with Claesson, Kitaev, and Ragnarsson. \par L'ordre de Bruhat munit tout groupe de Coxeter d'une structure de poset. L'idéal composé des éléments de ce poset engendrant des idéaux principaux ordonnés booléens, forme un poset simplicial. Ce poset simplicial définit le complexe booléen pour le groupe. Dans un système de Coxeter de rang n, nous montrons que le complexe booléen est homotopiquement équivalent à un bouquet de sphères de dimension (n-1). Le nombre de ces sphères est le nombre booléen, qui peut être calculé inductivement à partir du système de Coxeter non-étiquetté; définissant ainsi un invariant de graphe. Pour certaines familles de graphes, les nombres booléens satisfont des propriétés combinatoires intriguantes. Ce travail est une collaboration entre Claesson, Kitaev, et Ragnarsson.

Most balanced minimum cuts

We consider the problem of finding most balanced cuts among minimum s t -edge cuts and minimum s t -vertex cuts, for given vertices s and t , according to different balance criteria. For edge cuts [ S , S ¯ ] we seek to maximize min { | S | , | S ¯ | } . For vertex cuts C of G we consider the objectives of (i) maximizing min { | S | , | T | } , where { S , T } is a partition of V ( G ) ∖ C with s ∈ S , t ∈ T and [ S , T ] = 0̸ , (ii) minimizing the order of the largest component of G − C , and (iii) maximizing the order of the smallest component of G − C . All of these problems are NP-hard. We give a PTAS for the edge cut variant and for (i). These results also hold for directed graphs. We give a 2-approximation for (ii), and show that no non-trivial approximation exists for (iii) unless P=NP. To prove these results we show that we can partition the vertices of G , and define a partial order on the subsets of this partition, such that ideals of the partial order correspond bijectively to minimum s t -cuts of G . This shows that the problems are closely related to Uniform Partially Ordered Knapsack (UPOK), a variant of POK where element utilities are equal to element weights. Our algorithm is also a PTAS for special types of UPOK instances.

Obtainable sizes of topologies on finite sets

We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a range of obtainable sizes for such topologies.

Homotopy type of the boolean complex of a Coxeter system

In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, the boolean complex is homotopy equivalent to a wedge of ( n − 1 ) -dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group.

Promotion and Evacuation

Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schützenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.

Distributive lattices of small width, II: A problem from Stanley's 1986 text Enumerative Combinatorics

In Richard P. Stanley's 1986 text, Enumerative Combinatorics, the following problem is posed: Fix a natural number k. Consider the posets P of cardinality n such that, for 0 < i < n , P has exactly k order ideals (down-sets) of cardinality i. Let f k ( n ) be the number of such posets. What is the generating function ∑ f 3 ( n ) x n ? In this paper, the problem is solved.

Dialectical Displacement and Redeployment of Attachments in the Kenotic Mode of Conversion: The Leaving Behind of Anton Boisen in Psychosis

This paper introduces a kenotic theory of conversion that builds from simple attachment to childhood experience of peak states to encompass dialectical stages of development: Priming, Decentering, Reflection, Encounter, Denucleation, Emplacement and Discipline. Thereafter, the dialectical mode gives rise to the mature phase of conversion—the continuing integration of the religious worldview through Metamorphosis and Embassy. The conversion theory is illustrated by an interpretation of a dialectical set of experiences of Anton Boisen. I interpret Anton Boisen’s conversion from a 19th Century Christianity that was wedded to religious and racial manifest destiny to that of a reborn Christian living with the 20th Century’s experience of evangelical and evolutionary universalism. Boisen’s clinical desolations (“psychosis”) and adoption of vocation (“change of allegiance”) suggest his conversion counterposed the instinctual with the higher order ideals he struggled to embody—a dialectical negation of his younger static and triumphalist Christian cultural identity that developed into a more integrated, expansive, and inclusive view of the human family and deepening allegiance to the ordinary, underserved, and growing population of the mentally suffering.

Avoidable structures, II: Finite distributive lattices and nicely structured ordered sets

Let \(\langle {\mathcal{D}}, \leq \rangle\) be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We characterize the order ideals in \(\langle {\mathcal{D}}, \leq \rangle\) that are well-quasi-ordered by embeddability, and thus characterize the members of \(\mathcal{D}\) that belong to at least one infinite anti-chain in \(\mathcal{D}\).

Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras

For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems $A_{1}\oplus A_{1}$, $A_{2}$, $C_{2}$, and $G_{2}$. Examples appear in Figures Four and Five. We then form the distributive lattices of order ideals of these posets. Corollary CharacterCorollary gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem CharacterProposition describes how these lattices provide a new combinatorial setting for the Weyl characters of representations of rank two semisimple Lie algebras. Most of these lattices are new; the rest of them (or related structures) have arisen in the work of Stanley, Kashiwara, Nakashima, Littelmann, and Molev. In a future paper, one author shows that the posets constructed here form a Dynkin diagram-indexed answer to a combinatorially posed classification question. In a companion paper, some of these lattices are used to explicitly construct some representations of rank two semisimple Lie algebras. This implies that these lattices are strongly Sperner.

The poset perspective on alternating sign matrices

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

Liberty as progression: a study of the revolutions idealized in Areopagitica, The marriage of heaven and hell, and The matrix

I address the assumptions of Areopagitica, The Marriage of Heaven and Hell and The Matrix as well as provoke their ideals in order to expose the status of their liberty and progression towards truth as one that is in no sense absolute, but time and space specific.

Sala de aula e teceduras subjetivas

O propósito deste artigo é pensar a tarefa docente tendo como foco as vicissitudes a que a relação docente-discente está submetida e as quais desviam alunos e professores dos ideais de ordem e de harmonia. Com o objetivo de analisar os fluxos intersubjetivos em sala de aula, abordam-se aspectos da psicodinâmica grupal com o uso das noções psicanalíticas de grupo de suposto básico e de identificação projetiva. Com esse procedimento, pretende-se ao mesmo tempo sondar até que ponto noções eficazes na clínica psicanalítica podem ser úteis para analisar e elaborar situações pertencentes ao campo da Educação, em particular, à sala de aula concebida aqui como espaço de produção de intercorrências psíquicas. Procura-se pensar o espaço intersubjetivo segundo uma lógica que considere a criação de uma terceiridade no ensinar e aprender. Introduz-se a noção de "terceira voz" (Ogden) para indicar a criação e recriação do outro intersubjetivo, que faz da atividade docente uma experiência geradora de turbulência emocional com efeitos diversos nas constituições subjetivas. O enrijecimento identitário, com o concomitante desdém e intolerância à diferença, pode ser uma forma de reagir àquilo que escapa ao controle e à previsibilidade e que enfraquece desse modo os movimentos transformacionais implicados no processo de aprendizagem.

On averaging of rearrangement invariant ideals of the space L1 (Ω, Σ, μ) by non-atomic σ-subalgebras of Σ

The problem of averaging = into mapping) is studied for conditional expectation operators which are considered on rearrangement invariant order ideals of measurable functions given on non-atomic probability supports.

ON AN ORDER-BASED CONSTRUCTION OF A TOPOLOGICAL GROUPOID FROM AN INVERSE SEMIGROUP

AbstractWe show how to construct a topological groupoid directly from an inverse semigroup and prove that it is isomorphic to the universal groupoid introduced by Paterson. We then turn to a certain reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings), we prove that this reduction is the graph groupoid introduced by Kumjian \et (respectively, the tiling groupoid of Kellendonk). We also study the open invariant sets in the unit space of this reduction in terms of certain order ideals of the underlying inverse semigroup. This can be used to investigate the ideal structure of the associated reduced $C^\ast$-algebra.

On Whitney numbers of the order ideals of generalized fences and crowns

We solve some recurrences given by E. Munarini and N. Zagaglia Salvi proving explicit formulas for Whitney numbers of the distributive lattices of order ideals of the fence poset and crown poset. Moreover, we get a method to obtain explicit formulas for Whitney numbers of lattices of order ideals of fences with higher asymmetric peaks.

Urban Order and Street Regulation in Seventeenth-Century Sweden

AbstractThis article examines how, in the early modern towns of Stockholm and Åbo, royal interests, town planning, street building and maintenance, and street behavior related to ideas and ideals of urban order. Town laws and ordinances, royal letters and some town court records are employed to tell a story of royal interest in well-ordered, impressive, successful towns; various street plans for the capital and the smaller provincial towns; and the varying execution of renewal plans. It is evident that the capital was to reflect the royal person and the state and that streets and street behaviour were important in this regard. But in towns outside the capital, especially in concrete street maintenance, the centrality of streets does not clearly emerge. The burghers in towns operated as individuals—there was no bottom-up or top-down plan or supervision.

Stravinski et ses exégètes (1910-1940) by Valérie Dufour (review)

!52 SEER, 86, I, 2008 career, ensure his survival, and allow his prolific and poetic painting to achieve success, and through it to address the world. On the cover of thisbook Chagall's SelfPortraitwith SevenFingers shows the painter working fastwith Paris through thewindow but Russia in his mind. He signed inHebrew at the top and French-style lower right. When Chagall was latermade responsible for art in Soviet Vitebsk, he signed his Cubist Landscape (p. 16) again and again in a stream of letteringfalling like raindrops on a window pane, first in Hebrew then Russian, French and again in Russian, a list of his identities in his early life, in St Petersburg, in Paris and in Soviet Russia where he was deeply committed to the revival of Jewish Theatre. Many years later he was buried in a Christian cemetery at St-Paul-de-Vence in France. For Moyshe Shagal/Marc Chagall, as Harshav reveals it in this invaluable source book, the issue of identitywas never far away. Courtauld Instituteof Art,London John Milner Dufour, Val?rie. Stravinskiet ses ex?g?tes (igio-iQ4o). Facult? de Philosophie et Lettres, 112 ? Art. Editions de l'Universit? de Bruxelles, Brussels, 2006. 276 pp. Tables. Notes. Bibliography. Index. 24.00 (paperback). Stravinskii's love-hate relationship with the press and his despotic attitude towards his critics became inseparable aspects of his rigorous profile, a profile that reflected his aesthetic ideals of order, control and objectivity. It is this complex relationship and the interactionwith some of his critics thatDufour investigates in her book, focusing not on the reception, but on the exegesis of Stravinskii's output by various intellectuals during theyears 1910 to 1940.The former, she explains, involved occasional reviews of his music, while the latter refers to systematic attempts to grasp and interpret the Stravinskii 'phenom enon' based on each writer's personal set of ideas. Dufour studies texts by a number of 'exegetes' from several countries (Russia, USSR, France, Belgium, Germany, Italy and England), thus shedding light on their dialectical exchanges with the composer. No less than twenty cases of critics are explored whose connection and degree of proximity to Stravinskii varied substantially, ranging from opposition (Andrei Rimskii-Korsakov), to autonomy (Boris Asafev, Boris de Shletser), to close friendship and collaboration (Andr? Schaeffner, Charles-Albert Cingria, Petr Suvchinskii, Artur Lure). To be sure, Dufour is not the firstmusicologist to expose Stravinskii's relationship with his critics. In fact, the bottom line of her argument is his previously acknowledged desire for complete control over his image which, forDufour, renders him an incarnation of themedia epoch inwhich he lived. Yet, her study is a valuable contribution to understanding not only this specific issue, but also Stravinskii's neoclassicism more broadly. By analysing a vast number of contemporaneous articles and books ? many of them long overlooked ? by all thewriters she could correlate with Stravinskii, Dufour establishes thathis involvement in criticism of his production was much more systematic than formerly thought. On a second level, she demonstrates how REVIEWS IOS Stravinskii's interactionwith some of those intellectuals,many ofwhom acted as his propagandists, actually influenced his neoclassical aesthetics. She thus contextualizes and puts in perspective some of Stravinskii's neoclassical ideals and concepts. The strongest point of her research is undoubtedly the examina tion of hitherto unpublished archival sources, which she uses, for instance, to confirm Suvchinskii's key role in the composition of Stravinskii's Poetics of Music. Dufour does not hesitate to make some bold statements in her book. She claims, for example, that Robert Craft minimized the importance of Suvchinskii for Stravinskii, because the twomen played a very similar role in the composer's life (pp. 51, 57). She also makes some original associations, such as her interpretation of Lur'e's 'Neogothic and Neoclassic' as a response to Schoenberg's mocking Stravinskii's neoclassicism (pp. 93-97). However, on thewhole, the reader will probably feel overwhelmed by the amount of infor mation presented; at times, the book admittedly reads like a reference book. Dufour's investigation would have benefited from the examination of fewer case studies on those intellectualswho interactedmore closelywith Stravinskii, and a more thorough analysis of their texts and thought...

On extensions of ideals in posets

Two kinds of order ideals, namely, the n-prime ideals and k-closed ideals of a poset are introduced and shown to be equivalent in the sense that an order-ideal is ( n + 1 ) -prime if and only if it is n-closed. Also a prime radical theorem for these ideals is proved.

Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras

For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2 and 3.3. We then form the distributive lattices of order ideals of these posets. Corollary 5.4 gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem 5.3 describes how these lattices provide a new combinatorial setting for the Weyl characters of representations of rank two semisimple Lie algebras. Most of these lattices are new; the rest of them (or related structures) have arisen in work of Stanley, Kashiwara, Nakashima, Littelmann, and Molev. In a future paper, one author shows that the posets constructed here form a Dynkin diagram-indexed answer to a combinatorially posed classification question. In a companion paper, some of these lattices are used to explicitly construct some representations of rank two semisimple Lie algebras. This implies that these lattices are strongly Sperner.