Order Ideals Research Articles

Noncrossing partitions and Bruhat order

We prove that the restriction of Bruhat order to type A noncrossing partitions for the Coxeter element c=s1s2⋯sn defines a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the base change from the graphical basis of the Temperley–Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we introduce a new set of vectors counted by the Catalan numbers and give new bijections–fixing each reflection–between noncrossing partitions associated to distinct Coxeter elements.

The Rise of Modern Sport in Fin de Siècle São Paulo: Reading Elite and Bourgeois Sensibilities, the Popular Press, and the Creation of Cultural Capital

In the period between 1890 and 1910, British sport flourished in a rapidly modernizing Brazilian city. São Paulo boomed as a coffee export centre for the global market, growing into the second largest metropolis in Brazil. New businesses and industries developed and thousands of immigrants from around the world migrated to the expanding South American city. Along with the flow of new residents came new ideas, new attitudes, and new lifestyles. British sporting customs particularly attracted the attention of São Paulo’s wealthy elites and expanding middle classes who saw in these habits the potential to advertize their commitment to modern ideals of civilization and order. The new British-style sporting clubs that sprang up in São Paulo conferred the cultural capital that the leadership castes needed to gain and maintain their hegemony in the city’s rapidly changing social landscape. São Paulo’s press circulated these new sensibilities and revealed that the city’s sporting enthusiasts both reproduced Westernized norms and re-signified athletic sensibilities to fit Brazilian social patterns.

Iterative Properties of Birational Rowmotion II: Rectangles and Triangles

Birational rowmotion — a birational map associated to any finite poset $P$ — has been introduced by Einstein and Propp as a far-reaching generalization of the (well-studied) classical rowmotion map on the set of order ideals of $P$. Continuing our exploration of this birational rowmotion, we prove that it has order $p+q$ on the $\left( p, q\right) $-rectangle poset (i.e., on the product of a $p$-element chain with a $q$-element chain); we also compute its orders on some triangle-shaped posets. In all cases mentioned, it turns out to have finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the $AA$ case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets.

Pluralism and Dialectic: On James's Relation to Hegel

In this paper James's pluralism is examined in light of his critiques of ‘intellectualism’ and monistic idealism in order to elucidate his relationship to Hegel. Contrary to the strong anti-Hegelianism found throughout the writings of James, Hegel's dialectic and speculative logic are able to give a rational account of the continuity of objects and relations within experience that James struggled to articulate in A Pluralistic Universe. Neither James nor Hegel is an absolute pluralist or monist due to the interdependence of the concepts of unity and plurality, aptly described by Hegel in his Logic, and alluded to by James in various places throughout his work. Thus, the ambiguity of the nature of James's pluralism previously noted by scholars is explained, and the relevance of Hegel and dialectic for pragmatist theory is elucidated.

The toxic lining of men’s fashion consumption: The omnipresent force of hegemonic masculinity

Abstract While scholars have theorized about men’s motives for consuming fashion, few empirical studies have explored these theorizations against men’s lived experiences. A gap in knowledge exists between how scholars understand men’s fashion consumption and how men understand their fashion consumption. In this article, I examine men’s motives for consuming fashion based on individual interviews with 30 demographically diverse male consumers. Findings reveal that men are motivated to consume fashion because it provides benefits, including expressing identity, cultivating success and facilitating engagement. However, the pressures of body anxiety, exclusion and sartorial stress temper these positive outcomes. The common thread of hegemonic masculinity connect the various benefits and pressures that men experience when they engage in fashion. My research enhances knowledge about the operation of masculinity in consumer culture and the anxieties that men experience during the fashion consumption process. Menswear brands are advised to promote diverse masculine ideals in order to prevent negative business outcomes.

On modules of finite projective dimension

AbstractWe address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ringRof mixed characteristicp> 0, wherepis a nonzero divisor, ifIis an ideal of finite projective dimension overRandp𝜖Iorpis a nonzero divisor onR/I, then every minimal generator ofIis a nonzero divisor. Hence, ifPis a prime ideal of finite projective dimension in a local ringR, then every minimal generator ofPis a nonzero divisor inR.

On modules of finite projective dimension

AbstractWe address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ringRof mixed characteristicp> 0, wherepis a nonzero divisor, ifIis an ideal of finite projective dimension overRandp𝜖Iorpis a nonzero divisor onR/I, then every minimal generator ofIis a nonzero divisor. Hence, ifPis a prime ideal of finite projective dimension in a local ringR, then every minimal generator ofPis a nonzero divisor inR.

Toric partial orders

Toric partial orders correspond to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets arise from finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. There are natural toric analogues of many standard features of ordinary partial orders, such as chains, antichains, intervals, transitivity, Hasse diagrams, linear extensions, total orders, morphisms, and order ideals. Most of these only become apparent when one looks at these objects geometrically. Toric posets arise naturally in a wide variety of contexts, from the study of cyclic reducibility and conjugacy in Coxeter groups, to the critical path method (CPM) for scheduling activities with periodicity in operations research. This talk will be a survey on toric posets and it will be filled with lots of colorful pictures. This is joint work with Mike Develin and Vic Reiner.

Schools, Justice, and Immigrant Students: Assimilation, Race, Ethnicity, Gender, and Perceptions of Fairness and Order

Background/ContextStudents’ perceptions of justice, fairness, and order within their schools are arguably key building blocks of socialization to participation within a democratic society. The ideals of justice, fairness, and order within their schools are particularly imperative because the educational system is founded on a belief of democracy and meritocracy. It is also known that students’ perceptions of school justice can vary by race, ethnicity, and gender. What remains uncertain is how the fastest growing segment of the United States, students in immigrant families, perceive the school justice, fairness, and order within their school.PurposeThe aim of this study is to explore if straight-line assimilation, segmented assimilation, and immigrant optimism hypotheses explain the relationships between schools, justice, and immigration, as well as the potential role of gender, race, and ethnicity in immigrant youth perceptions of justice, fairness, and order.Participants/SubjectsThis study utilizes the Education Longitudinal Study of 2002 (ELS), a nationally representative sample of high school sophomores.Research DesignThis study's research design includes statistical analysis of secondary data.Findings/ResultsFindings do suggest that the students’ perceptions of justice, fairness, and order are indeed moderated by immigrant generation, race, ethnicity, and gender.Conclusions/RecommendationsEducators and educational researchers who are seeking to better understand the schooling experiences of immigrant youth might benefit from questioning assimilation and Americanization as processes that inevitably promote educational progress. Given that immigrant youth are and have historically been marginalized within U.S. schools, it appears that socialization, Americanization, gender, and immigrant generational status are germane to creating democratic education for all students. Attentiveness to democratic school justice, order, and fairness is, therefore, imperative.

Rhetoric Renouncing Rhetoric:

ABSTRACTThe problem St. Augustine confronts in the Confessions is fundamentally one of rhetoric: God should be singularly desirable, yet rhetoric seems necessary to motivate our pursuit of him. Religion participates in the relative marketplace of rhetoric, where ideals need to be authorized because they lack a self-sufficient rationale. In his early encounters with Cicero and the Platonists, Augustine struggles to renounce all such partial ideals in order to pursue philosophical truth unequivocally. Yet the refusal of rhetoric is, paradoxically, another willed ideal authorized by its own rhetoric. Augustine ultimately escapes rhetoric in the conversion scene by demonstrating his inescapable subjection to it; in doing so, he surrenders his will in such a way as to permit God's grace to operate through him. His conversion ultimately results from this inverted humiliation, which forces Augustine to abdicate his ascetic efforts and pretensions.

Eigenspace arrangements of reflection groups

The lattice of intersections of reflecting hyperplanes of a complex reflection group W W may be considered as the poset of 1 1 -eigenspaces of the elements of W W . In this paper we replace 1 1 with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset. After posing the main question of whether this poset is shellable, we show that all its upper intervals are geometric lattices and then answer the question in the affirmative for the infinite family G ( m , p , n ) G(m,p,n) of complex reflection groups, and the first 31 of the 34 exceptional groups, by constructing CL-shellings. In addition, we completely determine when these eigenspaces of W W form a K ( π , 1 ) K(\pi ,1) (resp. free) arrangement. For the symmetric group, we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing balanced partition posets, presented as particular upper order ideals of Dowling lattices, study the representation afforded by the top (co)homology group, and give a simple map to the posets of pointed d d -divisible partitions.

On the enumeration of [formula omitted]-core partitions

Anderson established a connection between core partitions and order ideals of certain posets by mapping a partition to its β-set. In this paper, we give a description of the posets P(s,s+1,s+2) whose order ideals correspond to (s,s+1,s+2)-core partitions. Using this description, we obtain the number of (s,s+1,s+2)-core partitions, the maximum size and the average size of an (s,s+1,s+2)-core partition, confirming three conjectures posed by Amdeberhan.

Lexicographic shellability, matroids, and pure order ideals

In 1977 Stanley conjectured that the h-vector of a matroid independence complex is a pure O-sequence. In this paper we use lexicographic shellability for matroids to motivate a new approach to proving Stanley's conjecture. This suggests that a pure O-sequence can be constructed from combinatorial data arising from the shelling. We then prove that our conjecture holds for matroids of rank at most four, settling the rank four case of Stanley's conjecture. In general, we prove that if our conjecture holds for all rank d matroids on at most 2d elements, then it holds for all matroids of rank d.

Measures on Cantor sets: The good, the ugly, the bad

We translate Akin’s notion of good (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups). This yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the related property of refinability, we consider goodness for sets of measures (traces on dimension groups), and obtain partial characterizations in terms of (special) convex subsets of Choquet simplices. These notions are also very closely related to unperforation of quotients of dimension groups by convex subgroups (that are not order ideals), and we give partial characterizations. Numerous examples illustrate the results.

Strategic taboos: chemical weapons and US foreign policy

This article examines US President Barack Obama's foreign policy rhetoric on Syria, specifically in relation to the threat of chemical weapons and the prohibitionary taboo surrounding their use. It contends that Obama's rhetorical construction of the taboo is not simply a commitment to the control of these horrific weapons (where such arms have been comprehended as so extensively vile as to preclude their employment), but that this also represents the strategic linguistic exploitation of these normative ideals in order to directly shape policy. By analysing of presidential speeches made during the conflict, it demonstrates that Obama has manipulated pre-existing conceptions of chemical weapons as taboo, and also as forms of weapons of mass destruction, to deliberately construct policy in line with his own political ambitions�most notably as a way of forcing a multilateral solution to the situation in Syria. This article challenges existing perceptions of the chemical weapons taboo as an inherently normative constraint, arguing that this instead comprises a more agency-driven construct. Static notions of the taboo must be abandoned and subsequently replaced with a framework of understanding that recognizes how the taboo can be used as a deliberate driver of foreign policy.

Conductor ideals of orders in algebraic number fields

We will give a new version of proof for Furtwangler’s characterization of all ideals of an algebraic number field which appear as conductors of orders of the field.

ISU TERORISME DAN RESPONS AKTIVIS MUDA ACEH

This article wants to examine the extent of radical thought in the minds of the younger generation pervasive in Aceh. With the new Aceh conditions experienced several years of post-conflict situation safe again struck by discovered a number of terrorists who make Jalin mountains forest, Aceh Besar, as a military training ground. Also found are also a number of books that have a narrative that can bring inspiration radicalism and terrorism acts. The results of in-depth interviews of young activists in Aceh, can be explained that the concept of terrorism models struggle can not be accepted by young activists in the city of Banda Aceh. They have to understand the teachings of Islam that is relatively comprehensive and proportional. But their ideals in order to uphold Islamic shariah in Aceh is a necessity.

Lattices from maximal orders into quaternion algebras

We propose an algebraic framework to construct dense lattices from maximal orders of a quaternion algebra whose center is an imaginary quadratic field. This work extends [21] where a method called algebraic reduction has been proposed to efficiently decode the Golden code [14]. In particular, we propose four new constructions of E8 lattice from left ideals of maximal orders of some quaternion algebras with centers Q(−d), d=1,2,3,7.

W. E. B. Du Bois and The souls of black folk

In this book, Stephanie J. Shaw brings a new understanding to one of the great documents of American and black history. While most scholarly discussions of The Souls of Black Folk focus on the veils, the color line, double consciousness, or Booker T. Washington, Shaw reads Du Bois' book as a profoundly nuanced interpretation of the souls of black Americans at the turn of the twentieth century. Demonstrating the importance of the work as a sociohistorical study of black life in America through the turn of the twentieth century and offering new ways of thinking about many of the topics introduced in Souls, Shaw charts Du Bois' successful appropriation of Hegelian idealism in order to add America, the nineteenth century, and black people to the historical narrative in Hegel's philosophy of history. Shaw adopts Du Bois' point of view to delve into the social, cultural, political, and intellectual milieus that helped to create The Souls of Black Folk.

Quasi-one-fibered ideals of order one in dimension two

Simple complete $\M$-primary ideals are fundamental in the Zariski-Lipman theory of complete ideals in a two-dimensional regular local ring $(R,\M)$. Complete $\M$-primary ideals of order one constitute a particular class of such ideals containing, for example, all the first neighborhood ideals of $R$. A number of interesting properties of complete $\M$-primary ideals of order one have been proved by several authors. For example, these ideals have only one Rees valuation (and hence are one-fibered) and they have the very simple form $(x_1^n,x_2)R$, $n\in\n_{+}$, with $x_1,x_2$ a minimal ideal basis of $\M$. In the present paper we investigate how far some results concerning these complete $\M$-primary ideals of order one can be extended to complete quasi-one-fibered $\M$-primary ideals of order one in a natural generalization of $R$: a two-dimensional normal Noetherian local domain with algebraically closed residue field and the associated graded ring an integrally closed domain.