Poset Structure Research Articles

Join operation for the Bruhat order and Verma modules

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type A. The statement is not true in other types, and we propose a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext space between a simple module and a Verma module. We give a conjectural complete description of such socles which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan–Lusztig polynomials.

Poset Structure Concerning Cylindric Diagrams

Cylindric diagrams admit the structure of infinite $d$-complete posets with natural ordering. The purpose of this paper is to provide a realization of a cylindric diagram as a subset of an affine root system of type A via colored hook lengths and to present several characterizations of its poset structure. Furthermore, the set of order ideals of a cylindric diagram is described as a weak Bruhat interval of the affine Weyl group.

Causal structure of spacetime and Scott topology

In this paper we establish the spacetime manifold as a partially ordered set via the casual structure. We show that these partially ordered sets are naturally continuous as a suitable way below relation can be established via the chronological order. We further consider those classes of spacetimes on which a lattice structure can be endowed by physically defining the joins and meets. By considering the physical properties of null geodesics on the spacetime manifold we show that these lattices are necessarily distributive. These lattices are then continuous as a result of the equivalence between the way below relation and chronology. This enables us to define the Scott topology on the spacetime manifold and describe it on an equal footing as any other continuous lattice. We further show that the Scott topology is a proper subset of Alexandroff topology, which must be the manifold topology for the strongly causal spacetimes, (and hence a coarser topology than Alexandroff). In the process we find some interesting results on the sobriety of these manifolds. We prove that they are necessarily not sober under the Scott topology but regain their sobriety under Alexandroff topology. We also define a dual Scott topology on these manifolds by endowing them with bicontinuous poset structure and show that the join of the Scott topology with the dual is the Alexandroff topology. We also discuss the previous works done in this topic and how the present work generalises those results to some extent.

Causal Order Complex and Magnitude Homotopy Type of Metric Spaces

Abstract In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space $X$ and a real parameter $\ell \geq 0$. This space is roughly consisting of all paths of length $\ell $ and has the reduced homology group that is isomorphic to the magnitude homology group of $X$. To construct the magnitude homotopy type, we consider the poset structure on the spacetime $X\times \mathbb{R}$ defined by causal (time- or light-like) relations. The magnitude homotopy type is defined as the quotient of the order complex of an intervals on $X\times \mathbb{R}$ by a certain subcomplex. The magnitude homotopy type gives a covariant functor from the category of metric spaces with $1$-Lipschitz maps to the category of pointed topological spaces. The magnitude homotopy type also has a “path integral” like expression for certain metric spaces. By applying discrete Morse theory to the magnitude homotopy type, we obtain a new proof of the Mayer–Vietoris-type theorem and several new results including the invariance of the magnitude under sycamore twist of finite metric spaces.

Homomorphism complexes, reconfiguration, and homotopy for directed graphs

The neighborhood complex of a graph was introduced by Lovász to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such ‘Hom complexes’ are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs G and H, we consider the polyhedral complex Hom⃗(G,H) that parametrizes the directed graph homomorphisms f:G→H. This construction can be seen as a special case of the poset structure on the set of multihomomorphisms in more general categories, as introduced by Kozlov, Matsushita, and others. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals.We study examples of directed Hom⃗ complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom⃗ complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom⃗ complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of Hom⃗(G,Tn) for Tn a tournament, obtaining a complete answer for the case of transitive Tn. If G=Tm is also a transitive tournament, we describe a connection to mixed subdivisions of dilated simplices. Finally we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom⃗ complexes.

MCRapper: Monte-Carlo Rademacher Averages for Poset Families and Approximate Pattern Mining

“I’m an MC still as honest” – Eminem, Rap God We present MCRapper , an algorithm for efficient computation of Monte-Carlo Empirical Rademacher Averages (MCERA) for families of functions exhibiting poset (e.g., lattice) structure, such as those that arise in many pattern mining tasks. The MCERA allows us to compute upper bounds to the maximum deviation of sample means from their expectations, thus it can be used to find both (1) statistically-significant functions (i.e., patterns) when the available data is seen as a sample from an unknown distribution, and (2) approximations of collections of high-expectation functions (e.g., frequent patterns) when the available data is a small sample from a large dataset. This flexibility offered by MCRapper is a big advantage over previously proposed solutions, which could only achieve one of the two. MCRapper uses upper bounds to the discrepancy of the functions to efficiently explore and prune the search space, a technique borrowed from pattern mining itself. To show the practical use of MCRapper , we employ it to develop an algorithm TFP-R for the task of True Frequent Pattern (TFP) mining, by appropriately computing approximations of the negative and positive borders of the collection of patterns of interest, which allow an effective pruning of the pattern space and the computation of strong bounds to the supremum deviation. TFP-R gives guarantees on the probability of including any false positives (precision) and exhibits higher statistical power (recall) than existing methods offering the same guarantees. We evaluate MCRapper and TFP-R and show that they outperform the state-of-the-art for their respective tasks.

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.

Diameter of the commutation classes graph of a permutation

We define a statistic on the graph of commutation classes of a permutation of the symmetric group which is used to show that these graphs are equipped with a ranked poset structure, with a minimum and maximum. This characterization also allows us to compute the diameter of the commutation graph for any permutation, from which the results for the longest permutation and for fully commutative permutations are recovered.

N∞-operads and associahedra

We provide a new combinatorial approach to studying the collection of N-infinity-operads in G-equivariant homotopy theory for G a finite cyclic group. In particular, we show that for G the cyclic group of order p^n the natural order on the collection of N-infinity-operads stands in bijection with the poset structure of the (n+1)-associahedron. We further provide a lower bound for the number of possible N-infinity-operads for any finite cyclic group G.

An Order on Circular Permutations

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

Fragments of Quasi-Nelson: The Algebraizable Core

Abstract This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic $FL_{ew}$ (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. $FL_{ew}$-algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.

On positive geometries of quartic interactions: one loop integrands from polytopes

Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas (AHST) [1] we explore the positive geometry encoding one loop scattering amplitude for quartic scalar interactions. We define a new class of combinatorial polytopes that we call pseudo-accordiohedra whose poset structures are associated to singularities of the one loop integrand associated to scalar quartic interactions. Pseudo-accordiohedra parametrize a family of projective forms on the abstract kinematic space defined by AHST and restriction of these forms to the type-D associahedra can be associated to one-loop integrands for quartic interactions. The restriction (of the projective form) can also be thought of as a canonical top form on certain geometric realisations of pseudo-accordiohedra. Our work explores a large class of geometric realisations of the type-D associahedra which include all the AHST realisations. These realisations are based on the pseudo-triangulation model for type-D cluster algebras discovered by Ceballos and Pilaud [2].

Magnitude homology of metric spaces and order complexes

Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order complexes of posets. In a metric space, an interval (the set of points between two chosen points) has a natural poset structure, which is called the interval poset. Under additional assumptions on sizes of $4$-cuts, we show that the magnitude chain complex can be constructed using tensor products, direct sums and degree shifts from order complexes of interval posets. We give several applications. First, we show the vanishing of higher magnitude homology groups for convex subsets of the Euclidean space. Second, magnitude homology groups carry the information about the diameter of a hole. Third, we construct a finite graph whose $3$rd magnitude homology group has torsion.

Largest acylindrical actions and Stability in hierarchically hyperbolic groups

We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, right-angled Coxeter groups, most 3 3 –manifold groups, right-angled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a “best” one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and right-angled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. In the appendix, it is verified that any space satisfying the a priori weaker property of being an “almost hierarchically hyperbolic space” is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.

L-homologies of double complexes

The notion of L-homologies (of double complexes) as proposed in this paper extends the notion of classical horizontal and vertical homologies along with two other new homologies introduced in the homological diagram lemma called the salamander lemma. We enumerate all L-homologies associated with an object of a double complex and provide new examples of exact sequences. We describe a classification problem of these exact sequences. We study two poset structures on these L-homologies; one of them determines the trivialities of horizontal and vertical homologies of an object in terms of other L-homologies of that object, whereas the second structure shows the significance of the two homologies introduced in the salamander lemma. Finally, we prove the existence of a faithful amnestic Grothendieck fibration from the category of L-homologies to a category consisting of objects and morphisms of a given double complex.

Toric Bruhat interval polytopes

For two elements v and w of the symmetric group Sn with v≤w in Bruhat order, the Bruhat interval polytope Qv,w is the convex hull of the points (z(1),…,z(n))∈Rn with v≤z≤w. It is known that the Bruhat interval polytope Qv,w is the moment map image of the Richardson variety Xw−1v−1. We say that Qv,w is toric if the corresponding Richardson variety Xw−1v−1 is a toric variety. We show that when Qv,w is toric, its combinatorial type is determined by the poset structure of the Bruhat interval [v,w] while this is not true unless Qv,w is toric. We are concerned with the problem of when Qv,w is (combinatorially equivalent to) a cube because Qv,w is a cube if and only if Xw−1v−1 is a smooth toric variety. We show that a Bruhat interval polytope Qv,w is a cube if and only if Qv,w is toric and the Bruhat interval [v,w] is a Boolean algebra. We also give several sufficient conditions on v and w for Qv,w to be a cube.

The A-fibered Burnside Ring as A-Fibered Biset Functor in Characteristic Zero

Let A be an abelian group such that torn(A) is finite for every n ≥ 1 and let ${\mathbb{K}}$ be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in A. In this paper we prove fundamental properties of the A-fibered Burnside ring functor $B_{\mathbb{K}}^{A}$ as an A-fibered biset functor over K. This includes a description of the composition factors of $B^{A}_{\mathbb{K}}$ and the lattice of subfunctors of $B_{\mathbb{K}}^{A}$ in terms of what we call BA-pairs and a poset structure on their isomorphism classes. Unfortunately, we are not able to classify BA-pairs. The results of the paper extend results of Coskun and Yilmaz for the A-fibered Burnside ring functor restricted to p-groups and results of Bouc in the case that A is trivial, i.e., the case of the Burnside ring functor as a biset functor over fields of characteristic zero. In the latter case, BA-pairs become Bouc’s B-groups which are also not known in general.

A Compact Representation for Modular Semilattices and Its Applications

A modular semilattice is a semilattice generalization of a modular lattice. We establish a Birkhoff-type representation theorem for modular semilattices, which says that every modular semilattice is isomorphic to the family of ideals in a certain poset with additional relations. This new poset structure, which we axiomatize in this paper, is called a PPIP (projective poset with inconsistent pairs). A PPIP is a common generalization of a PIP (poset with inconsistent pairs) and a projective ordered space. The former was introduced by Barthelemy and Constantin for establishing Birkhoff-type theorem for median semilattices, and the latter by Herrmann, Pickering, and Roddy for modular lattices. We show the Θ(n) representation complexity and a construction algorithm for PPIP-representations of (∧,∨)-closed sets in the product Ln of modular semilattice L. This generalizes the results of Hirai and Oki for a special median semilattice Sk. We also investigate implicational bases for modular semilattices. Extending earlier results of Wild and Herrmann for modular lattices, we determine optimal implicational bases and develop a polynomial time recognition algorithm for modular semilattices. These results can be applied to retain the minimizer set of a submodular function on a modular semilattice.

Lambda -Semidirect products of inverse monoids are weakly Schreier extensions

A split extension of monoids with kernel k :N rightarrow G, cokernel e :G rightarrow H and splitting s :H rightarrow G is weakly Schreier if each element g in G can be written g = k(n)se(g) for some n in N. The characterization of weakly Schreier extensions allows them to be viewed as something akin to a weak semidirect product. The motivating examples of such extensions are the Artin glueings of topological spaces and, of course, the Schreier extensions of monoids which they generalise. In this paper we show that the lambda -semidirect products of inverse monoids are also examples of weakly Schreier extensions. The characterization of weakly Schreier extensions sheds some light on the structure of lambda -semidirect products. The set of weakly Schreier extensions between two monoids comes equipped with a natural poset structure, which induces an order on the set of lambda -semidirect products between two inverse monoids. We show that Artin glueings are in fact lambda -semidirect products and inspired by this identify a class of Artin-like lambda -semidirect products. We show that joins exist for this special class of lambda -semidirect product in the aforementioned order.

Tableau posets and the fake degrees of coinvariant algebras

We introduce two new partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. Both posets are ranked by the major index statistic offset by a fixed shift. The existence of such ranked poset structures allows us to classify the realizable major index statistics on standard tableaux of arbitrary straight shape and certain skew shapes. By a theorem of Lusztig–Stanley, this classification can be interpreted as determining which irreducible representations of the symmetric group exist in which homogeneous components of the corresponding coinvariant algebra, strengthening a recent result of the third author for the modular major index. Our approach is to identify patterns in standard tableaux that allow one to mutate descent sets in a controlled manner. By work of Lusztig and Stembridge, the arguments extend to a classification of all nonzero fake degrees of coinvariant algebras for finite complex reflection groups in the infinite family of Shephard–Todd groups.