Arbitrary Poset Research Articles

Relative poset polytopes and semitoric degenerations

The two best studied toric degenerations of the flag variety are those given by the Gelfand–Tsetlin and FFLV polytopes. Each of them degenerates further into a particular monomial variety which raises the problem of describing the degenerations intermediate between the toric and the monomial ones. Using a theorem of Zhu one may show that every such degeneration is semitoric with irreducible components given by a regular subdivision of the corresponding polytope. This leads one to study the parts that appear in such subdivisions as well as the associated toric varieties. It turns out that these parts lie in a certain new family of poset polytopes which we term relative poset polytopes: each is given by a poset and a weakening of its order relation. In this paper we give an in depth study of (both common and marked) relative poset polytopes and their toric varieties in the generality of an arbitrary poset. We then apply these results to degenerations of flag varieties. We also show that our family of polytopes generalizes the family studied in a series of papers by Fang, Fourier, Litza and Pegel while sharing their key combinatorial properties such as pairwise Ehrhart-equivalence and Minkowski-additivity.

Poisson structures on finitary incidence algebras

We give a full description of the Poisson structures on the finitary incidence algebra FI(P,R) of an arbitrary poset P over a commutative unital ring R.

The monoid of monotone functions on a poset and quasi-arithmetic multiplicities for uniform matroids

We describe the structure of the monoid of natural-valued monotone functions on an arbitrary poset. For this monoid we provide a presentation, a characterization of prime elements, and a description of its convex hull. We also study the associated monoid ring, proving that it is normal, and thus Cohen-Macaulay. We determine its Cohen-Macaulay type, characterize the Gorenstein property, and provide a Gröbner basis of the defining ideal. Then we apply these results to the monoid of quasi-arithmetic multiplicities on a uniform matroid. Finally we state some conjectures on the number of irreducibles for the monoid of multiplicities on an arbitrary matroid.

Globally Hyperbolic Spacetimes as Posets

It is well-known that a spacetime with its causal relation is a partially ordered set (poset for short). If it is globally hyperbolic, then it is a bicontinuous poset whose the interval topology is the manifold topology. In this work, we will state a new condition on a poset, which is called DS-FI cluster point condition and we show that when a causally simple spacetime ${\mathscr{M}}$ as a poset with the manifold topology satisfies this condition, then ≪= I+ (where by ≪, we mean the way-below relation on arbitrary poset and by I+, we mean the chronological relation on spacetime), and ${\mathscr{M}}$ is a bicontinuous poset whose the interval topology is the manifold topology. Furthermore, we show that, on a causally simple spacetime ${\mathscr{M}}$ as a poset, the global hyperbolicity condition is strictly stronger than the DS-FI cluster point condition.

Investigating posets via their maximal chains

By considering the number of maximal chains going through each element of an arbitrary poset, we prove an extension of Erdős’s generalisation of Sperner’s Theorem, together with a partial converse. By considering the number of maximal chains between pairs of comparable elements, we also prove a generalisation of the LYM inequality.

L-Fuzzy Filters of a Poset

Many generalizations of ideals and filters of a lattice to an arbitrary poset have been studied by different scholars. The authors of this paper introduced several generalizations of L-fuzzy ideal of a lattice to an arbitrary poset in [1]. In this paper, we introduce several L-fuzzy filters of a poset which generalize the L-fuzzy filter of a lattice and give several characterizations of them.

Semimodularity and the Jordan–Hölder theorem in posets, with applications to partial partitions

Lattice-theoretical generalizations of the Jordan–Holder theorem of group theory give isomorphisms between finite maximal chains with same endpoints. The best one has been given by Czedli and Schmidt (after Gratzer and Nation), and it applies to semimodular lattices and gives a chain isomorphism by iterating up and down the perspectivity relation between intervals $$[x\wedge y,x]$$ and $$[y,x\vee y]$$ where x covers $$x\wedge y$$ and $$x\vee y$$ covers y. In this paper, we extend to arbitrary (and possibly infinite) posets the definitions of standard semimodularity and of the slightly weaker “Birkhoff condition”, following the approach of Ore (Bull Amer Math Soc 49(8):567–568, 1943). Instead of perspectivity, we associate tags to the covering relation, a more flexible approach. We study the finiteness and length constancy of maximal chains under both conditions and obtain Jordan–Holder theorems. Our theory is easily applied to groups, to closure ranges of an arbitrary poset, and also to five new order relations on the set of partial partitions of a set (i.e. partitions of its subsets), which do not constitute lattices.

Using TPA to count linear extensions

A linear extension of a poset P is a permutation of the elements of the set that respects the partial order. Let L(P) be the set of linear extensions. It is a #P complete problem to determine the size of L(P) exactly for an arbitrary poset, and so randomized approximation algorithms are used that employ samples that are uniformly drawn from the set of linear extensions. In this work, a new randomized approximation algorithm is proposed where the linear extensions are not uniformly drawn, but instead come from a distribution indexed by a continuous parameter β. The introduction of β allows for the use of a more efficient method for approximating #L(P) called TPA. Our primary result is that it is possible to sample from this continuous embedding in time that as fast or faster than the best known methods for sampling uniformly from linear extensions. For a poset containing n elements, this means we can approximate L(P) to within a factor of 1+ϵ with probability at least 1−δ using O(n3(ln⁡n)(ln⁡#L(P))2ϵ−2ln⁡δ−1) expected number of random uniforms and comparisons.

The L-lim-inf-convergence in fuzzy posets

In domain theory, the lim-inf-convergence in posets was introduce to characterize continuous posets. In this paper, the concept of the L-lim-inf-convergence of nets (xi)i∈I in fuzzy posets is proposed and its relationship with continuous fuzzy posets is studied. It is shown that for an arbitrary fuzzy poset, the L-lim-inf-convergence is topological if and only if the fuzzy poset is continuous.

POSETS ADMITTING THE LINEARITY OF ISOMETRIES

Abstract. In this paper, we deal with a characterization of the posetswith the property that every poset isometry of F nq fixing the origin is alinear map. We say such a poset to be admitting the linearity of isome-tries. We show that a poset P admits the linearity of isometries over F nq if and only if P is a disjoint sum of chains of cardinality 2 or 1 whenq= 2, or P is an anti-chain otherwise. 1. IntroductionLet F q be a finite field with q elements, and F nq the vector space of n-tuplesover F q . In 1995, Brualdi et al. [1] introduced a non-Hamming metric on F n which is associated to an arbitrary poset on [n] = {1,2,...,n}. It is calleda poset metric. The poset metric spaces have been extensively studied in[1, 3, 4, 5, 8]. We briefly introduce basic notions for poset metric on F nq .Let P = ([n],≤) be a poset on [n] of coordinate positions of vectors on F nq .A subset I of P is called an order ideal (or a down-set) if x ∈ I and y ≤ ximply y ∈ I. For an arbitrary subset A of P, we denote by hAi the smallestorder ideal of P containing A. The P-weight of a vector u = (u

The $m$-Cover Posets and the Strip-Decomposition of $m$-Dyck Paths

In the first part of this article we present a realization of the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$ in terms of $m$-tuples of Dyck paths of height $n$, equipped with componentwise rotation order. For that, we define the $m$-cover poset $\mathcal{P}^{\langle m \rangle}$ of an arbitrary bounded poset $\mathcal{P}$, and show that the smallest lattice completion of the $m$-cover poset of the Tamari lattice $\mathcal{T}_n$ is isomorphic to the $m$-Tamari lattice $\mathcal{T}_n^{(m)}$. A crucial tool for the proof of this isomorphism is a decomposition of $m$-Dyck paths into $m$-tuples of classical Dyck paths, which we call the strip-decomposition. Subsequently, we characterize the cases where the $m$-cover poset of an arbitrary poset is a lattice. Finally, we show that the $m$-cover poset of the Cambrian lattice of the dihedral group is a trim lattice with cardinality equal to the generalized Fuss-Catalan number of the dihedral group. Dans la première partie de cet article nous présentons une réalisation du treillis $m$ -Tamari $\mathcal{T}_n^{(m)}$ à l’aide de $m$-uplets de chemins de Dyck de hauteur $n$, équipés de l’ordre de rotation composante par composante. Pour cela, nous définissons le poset de $m$-couverture $\mathcal{P}^{\langle m \rangle}$ d’un poset borné quelconque $\mathcal{P}$, et montrons que la plus petite complétion en treillis du poset de $m$-couverture du treillis de Tamari $\mathcal{T}_n$ est isomorphe au treillis $m$-Tamari $\mathcal{T}_n^{(m)}$. Unoutil crucial pour la preuve de cet isomorphisme est une décomposition des chemins $m$-Dyck en $m$-uplets de chemins de Dyck usuels, que nous appelons la décomposition en bandes. Par la suite, nous caractérisons les cas où le poset de $m$-couverture d’un poset donné est un treillis. Enfin nous montrons que le poset de $m$-couverture du treillis Cambrien du groupe diédral est un treillis svelte de cardinalité le nombre généralisé de Fuss-Catalan du groupe diédral.

Anti-automorphisms and involutions on (finitary) incidence algebras

In the first part of this article, we show that the finitary incidence algebra of an arbitrary poset X over a field K has an anti-automorphism (involution) if and only if X has an anti-automorphism (involution), which generalizes the Theorem 2 of Spiegel [E. Spiegel, Involutions in incidence algebras, Linear Algebra Appl. 405 (2005), pp. 155–162]. In the second part of this article we generalize the decomposition theorem of Brusamarello and Lewis [R. Brusamarello and D.W. Lewis, Automorphisms and involutions on incidence algebras, Linear Multilinear Algebra (to appear)] and the results on classification of involutions of Brusamarello et al. [R. Brusamarello, E.Z. Fornaroli, and E.A. Santulo Jr, Classification of involutions on incidence algebras, Comm. Algebra (to appear)] to the case when X is locally finite.

Finitary Incidence Algebras

It is well known that incidence algebras can be defined only for locally finite partially ordered sets (Doubilet et al., 1972; Stanley 1986). At the same time, for example, the poset of cells of a noncompact cell partition of a topological space is not locally finite. On the other hand, some operations, such as the order sum and the order product (Stanley, 1986), do not save the locally finiteness. So it is natural to try to generalize the concept of incidence algebra. In this article, we consider the functions in two variables on an arbitrary poset (finitary series), for which the convolution operation is defined. We obtain the generalization of incidence algebra—finitary incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained.

Ordering the order of a distributive lattice by itself

We order the ordering relation of an arbitrary poset P component-wise by itself, obtaining a poset Φ( P) extending P. In particular, the effects of Φ on L ∈ DLAT01, the category of all bounded distributive lattices, are studied, mainly with the aid of Priestley duality. We characterize those L ∈ DLAT01 which occur as Φ( K) for some K ∈ DLAT01, decide this situation in polynomial time for finite L, characterize fixpoints of Φ within DLAT01 and relate them to free objects in DLAT01.

Dedekind–MacNeille Completion of n-ordered Sets

A completion of an n-ordered set \({\mathbf{P}}{\text{ = }}{\left\langle {P, \lesssim _{1} , \ldots , \lesssim _{n} } \right\rangle }\) is defined, by analogy with the case of posets (2-ordered sets), as a pair \({\left\langle {e,{\mathbf{Q}}} \right\rangle }\), where Q is a complete n-lattice and \(e{\text{:}}{\mathbf{P}} \to {\mathbf{Q}}\) is an n-order embedding. The Basic Theorem of Polyadic Concept Analysis is exploited to construct a completion of an arbitrary n-ordered set. The completion reduces to the Dedekind–MacNeille completion in the dyadic case, the case of posets. A characterization theorem is provided, analogous to the well-known dyadic one, for the case of joined n-ordered sets. The condition of joinedness is trivial in the dyadic case and, therefore, this characterization theorem generalizes the uniqueness theorem for the Dedekind–MacNeille completion of an arbitrary poset.

Monotone Nonincreasing Random Fields on Partially Ordered Sets. I

For an arbitrary poset H and measure ρ on H × R (where R is the real axis), we construct a monotone decreasing stochastic field ηρ and compute its finite-dimensional distributions. In the case where H is a Λ-semilattice and the measure ρ satisfies additional conditions, we compute various characteristics of the field ηρ such as the expectation of the field value at a point, variance of the field value at a point, and correlation function of the field. The described construction of random fields gives a new method for constructing positive definite functions on posets. Bibliography: 6 titles.

EMSO-logic and automata related to homogeneous flow event structures

In this paper we study the recognizability and definability on existential monadic second order (EMSO) logic of languages related to homogeneous flow event structures (HFES), i.e., the languages of configurations (posets) and the languages of proving sequences (words). An HFES is specified by an arbitrary poset G = (V,⩽) and a finite FES F as a result of copying F in vertices of G with additional homogeneous flow and conflict relations on the events of adjacent vertices. The problem is: whether the sets of configurations of HFES and languages of proving sequences are EMSO-definable. Results refer to the domains of linear orders, binary trees, and two-dimensional grids. If G is the domain of linear order or binary tree then the recognizability and EMSO-definability of configurations of HFES hold. We give some classes of HFES where the recognizability does not hold for the two-dimensional grid. The EMSO-definability of proving sequences is proved for all considered classes of HFES.

Formal topologies on the set of first-order formulae

The completeness proof for first-order logic by Rasiowa and Sikorski [13] is a simplification of Henkin's proof [7] in that it avoids the addition of infinitely many new individual constants. Instead they show that each consistent set of formulae can be extended to a maximally consistent set, satisfying the following existence property: if it contains (∃x)ϕit also contains some substitutionϕ(y/x) of a variableyforx. In Feferman's review [5] of [13], an improvement, due to Tarski, is given by which the proof gets a simple algebraic form.Sambin [16] used the same method in the setting of formal topology [15], thereby obtaining a constructive completeness proof. This proof is elementary and can be seen as a constructive and predicative version of the one in Feferman's review. It is a typical, and simple, example where the use of formal topology gives constructive sense to the existence of a generic object, satisfying some forcing conditions; in this case an ultrafilter satisfying the existence property.In order to get a formal topology on the set of first-order formulae, Sambin used the Dedekind-MacNeille completion to define a covering relation ⊲DM. This method, by which an arbitrary poset can be extended to a complete poset, was introduced by MacNeille [9] and is a generalization of the construction of real numbers from rationals by Dedekind cuts. It is also possible to define an inductive cover, ⊲I, on the set of formulae, which can also be used to give canonical models, see Coquand and Smith [3].

Weakly Purely Finitely Additive Measures

AbstractLet L be an orthomodular poset. A positive measure ξ on L is said to be weakly purely finitely additive if the zero measure is the only completely additive measure majorized by ξ. It was shown in [15] that, in an arbitrary orthomodular poset L, every positive measures μ is the sum v + ξ of a positive completely additive measure v and a weakly purely finitely additive measure ξ. We give sufficient conditions for this Yosida-Hewitt-type decomposition to be unique.A positive measure λ on L is said to be filtering if every non-zero element p in L majorizes a non-zero element q on which λ vanishes. A filtering measure is weakly purely finitely additive. Filtering measures play a mediator role throughout these investigations since some of the aforementioned conditions are given in terms of these.The results obtained here are then viewed in the context of Boolean lattices and applied to lattices of idempotents of non-associative JBW-algebras.

Hamilton Paths in Graphs of Linear Extensions for Unions of Posets

This paper proves that if a poset Q has an even number of linear extensions and these extensions can be generated by adjacent transpositions, then linear extensions of union of poset Q and an arbitrary poset P can also be generated by adjacent transpositions. This result is then applied to posets P and Q, which are sums of disjoint chains.