Maximal Chains Research Articles

Poly-antimatroid polyhedra

The notion of “antimatroid with repetition” was conceived by Bjorner, Lovasz and Shor in 1991 as an extension of the notion of antimatroid in the framework of non-simple languages. Further they were investigated by the name of “poly-antimatroids” (Nakamura, 2005, Kempner & Levit, 2007), where the set system approach was used. If the underlying set of a poly-antimatroid consists of n elements, then the poly-antimatroid may be represented as a subset of the integer lattice Z n . We concentrate on geometrical properties of two-dimensional ( n = 2 ) poly-antimatroids - poly-antimatroid polygons, and prove that these polygons are parallelogram polyominoes. We also show that each two-dimensional poly-antimatroid is a poset poly-antimatroid, i.e., it is closed under intersection. The convex dimension c d i m ( S ) of a poly-antimatroid S is the minimum number of maximal chains needed to realize S . While the convex dimension of an n -dimensional poly-antimatroid may be arbitrarily large, we prove that the convex dimension of an n -dimensional poset poly-antimatroid is equal to n .

Causes and (in)Determinism

The five papers of the current issue originate from selected talks of a workshop Causes and Tenses: Formal Perspectives, held in Krakow (Poland) in September 2010. The aim of this event was to explore two relatively recent projects in formal philosophy: a rigorous analysis of a modal and tensed notion of (in)determinism and a study of common causation by means of probabilistic extendability. In contrast to the Laplacean (in)determinism framed in tenseless terms of laws of nature or models of a theory, the first project sets out the task of analyzing a notion of (in)determinism exhibited in claims like ‘‘It is not settled yet if there will be a sea battle tomorrow; however, by tomorrow evening it will have been settled whether or not there was the battle’’. The analysis draws on the insights of the PriorThomason tense logic as well as Kaplan’s logic of indexicals. Beginning with Belnap’s (1992) theory of branching space-times, another aim of the project is to relate tensed (in)determinism to spatiotemporal structures of current physics. The present issue begins with three papers on some aspects of this variety of (in)determinism. In the first paper Thomas Muller reflects on the notion of possible history, which is employed in all schools emerging from the Prior-Thomason theory of branching time. A history plays a distinctive role in assessing the truth values of the sentences to be evaluated in a given model. Possible histories also offer a theoretical grip on the concept of (in)determinism. They are usually defined as set-theoretically maximal objects of a certain kind. Although the formal details depend on the particular theory, the common idea is to read this maximality as maximality with respect to modal consistency. This calls in turn for spelling out a criterion of modal consistency (historicity) which would accord with insights of physics. The existing criteria, identifying possible histories with maximal chains or maximal upward directed subsets of a base set, are, however, in conflict with some relativistic

The smallest regular polytopes of given rank

An abstract polytope is called regular if its automorphism group has a single orbit on flags (maximal chains). In this paper, the regular n-polytopes with the smallest number of flags are found, for every rank n>1. With a few small exceptions, the smallest regular n-polytopes come from a family of ‘tight’ polytopes with 2⋅4n−1 flags, one for each n, with Schläfli symbol {4∣4∣⋯∣4}. Also with few exceptions, these have both the smallest number of elements, and the smallest number of edges in their Hasse diagram.

Chain rotations: A new look at tree distance

As well known the rotation distance D(S,T) between two binary trees S, T of n vertices is the minimum number of rotations of pairs of vertices to transform S into T. We introduce the new operation of chain rotation on a tree, involving two chains of vertices, that requires changing exactly three pointers in the data structure as for a standard rotation, and define the corresponding chain distanceC(S,T). As for D(S,T), no polynomial time algorithm to compute C(S,T) is known. We prove a constructive upper bound and an analytical lower bound on C(S,T) based on the number of maximal chains in the two trees. More precisely we prove the general upper bound C(S,T)⩽n−1 and we show that there are pairs of trees for which this bound is tight. No similar result is known for D(S,T) where the best upper and lower bounds are 2n−6 and 53n−4 respectively.

Note on the [formula omitted]-vectors of cutsets in the subspace lattice

A cutset in the subspace lattice ℒn(q) (i.e., the poset of all subspaces of an n-dimensional vector space Fqn over the finite field Fq with q elements, ordered by inclusion) is a subset of ℒn(q) that intersects every maximal chain. We find a cutset in ℒn(q) that contains a fixed percentage α (0<α≤1) of the subspaces of each possible dimension. An asymptotic estimate to the greatest lower bound of such α (denoted by α(n)) is established. In particular, limn→∞α(n)=0.

Lie algebras acting semitransitively

A set S of linear operators on a vector space acts semitransitively if, given nonzero vectors x,y, there exists an operator a∈S such that either ax=y or ay=x. We show that for a Lie algebra g acting on a finite-dimensional complex vector space X this is equivalent to the existence of a (necessarily unique) maximal chain 0=Y0<Y1<⋯<Yn=X of g-invariant subspaces such that the orbit gx, x∈X, coincides with the smallest Yi containing x. In particular, a Lie algebra acting irreducibly and semitransitively actually acts transitively. We also show that every Lie algebra acting semitransitively contains a solvable subalgebra with the same property.

Minimal non-totally smooth groups

A group G is called smooth if its subgroup lattice L(G) has a smooth chain, and we call G totally smooth if all maximal chains in L(G) are smooth. We call G is a minimal non-totally smooth group if G is not totally smooth with totally smooth proper subgroups. In this paper we determine all finite minimal non-totally smooth groups. Mathematics Subject Classification: 20D30, 20E15

On the number of maximal chain transitive sets in fiber bundles

This article studies chain transitivity for semigroup actions on fiber bundles whose typical fibers are compact topological spaces. We discuss the number of maximal chain transitive sets and, as a consequence, we obtain conditions for the existence of a finest Morse decomposition. Some of the results obtained are applied to orthonormal frame and Stiefel manifolds. A description of the maximal chain transitive sets is provided in terms of the action of shadowing semigroups. This description is well known in the literature under the hypothesis of local transitivity. Here, we exclude the hypothesis of local transitivity when the state space is a compact quotient space of a topological group.

Maximal Chains of Copies of the Rational Line

We show that the class of order types of maximal chains in the partial order \(\langle E(\mathbb Q)\cup \{\emptyset \}, \subset \rangle\), where \(E(\mathbb Q)\) is the set of all subsets X of the rational line, \({\mathbb Q}\), such that X with the inherited order is isomorphic to \({\mathbb Q}\), is exactly the class of order types of compact sets of reals having the minimum non-isolated.

A Note on the ( FMC ) Condition for Extensions of Commutative Rings

A ring extension R ⊂ S is said to satisfy the (FMC) condition if there is a finite maximal chain of rings from R to S (cf. [1]). The aim of this note is to prove that if R ⊂ S is an extension of integral domains satisfying the (FMC) condition, then S is a P -extension of R. As a consequence, we establish that the polynomial extension R[X] ⊂ S[X] does never satisfy the (FMC) condition.

Bounds of the number of leaves of spanning trees

We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\over 4}(s-2)+2$ leaves. Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph $G$ does not exceed $k\ge 1$. We prove that $G$ has a spanning tree with at least $\alpha_{g,k}(v(G)-k-2)+2$ leaves, where $\alpha_{g,k}= {[{g+1\over2}]\over [{g+1\over2}](k+3)+1}$ for $k<g-2$; $\alpha_{g,k}= {g-2\over (g-1)(k+2)}$ for $k\ge g-2$. We present infinite series of examples showing that all these bounds are exact.

Lattice polyhedra and submodular flows

Lattice polyhedra, as introduced by Gröflin and Hoffman, form a common framework for various discrete optimization problems. They are specified by a lattice structure on the underlying matrix satisfying certain sub- and supermodularity constraints. Lattice polyhedra provide one of the most general frameworks of total dual integral systems. So far no combinatorial algorithm has been found for the corresponding linear optimization problem. We show that the important class of lattice polyhedra in which the underlying lattice is of modular characteristic can be reduced to the Edmonds–Giles polyhedra. Thus, submodular flow algorithms can be applied to this class of lattice polyhedra. In contrast to a previous result of Schrijver, we do not explicitly require that the lattice is distributive. Moreover, our reduction is very simple in that it only uses an arbitrary maximal chain in the lattice.

Notes on CD-independent subsets

It is proved in [8] that any two CD-bases in a finite distributive lattice have the same number of elements. We investigate CD-bases in posets, semilattices and lattices. It is shown that their CD-bases can be characterized as maximal chains in a related poset or lattice. We point out two known lattice classes characterized by some “0-conditions” whose CD-bases satisfy the mentioned property.

Computing maximal chains

In 1967 Wolk proved that every well partial order (wpo) has a maximal chain; that is a chain of maximal order type. (Note that all chains in a wpo are well-ordered.) We prove that such maximal chain cannot be found computably, not even hyperarithmetically: No hyperarithmetic set can compute maximal chains in all computable wpos. However, we prove that almost every set, in the sense of category, can compute maximal chains in all computable wpos. Wolk's original result actually shows that every wpo has a strongly maximal chain, which we define below. We show that a set computes strongly maximal chains in all computable wpo if and only if it computes all hyperarithmetic sets.

On Finite Maximal Chains of Weak Baer Going-Down Rings

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ⊆ T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T.

Commutative algebra of statistical ranking

A model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items. We represent the orderings as maximal chains in a graded poset. The most widely used ranking models are parameterized by rational function in the model parameters, so they define algebraic varieties. We study these varieties from the perspective of combinatorial commutative algebra. One of our models, the Plackett–Luce model, is non-toric. Five others are toric: the Birkhoff model, the ascending model, the Csiszár model, the inversion model, and the Bradley–Terry model. For these models we examine the toric algebra, its lattice polytope, and its Markov basis.

On calculating residuated approximations

For an order-preserving map f : L → Q between two complete lattices L and Q, there exists a largest residuated map ρf under f, which is called the residuated approximation of f. Andreka, Greechie, and Strecker introduced the notion of the shadow σf of f Iterations of the shadow are called the umbral mappings. The umbral mappings form a decreasing net that converges to the residuated approximation ρf of f. The umbral number uf of f is the smallest ordinal number α such that the equation \({\sigma^{(\alpha)}_{f} = \rho_{f}}\) holds. In order to speed up the computation of the umbral number uf of f and find some relation between the structure of L and uf, we present the concept of the order skeleton of a lattice \({L, \tilde{L} = L/\sim}\), determined by a certain congruence relation ~ on L where each equivalence class [x] is the maximal autonomous chain containing x. If [x] is finite for each \({x \in L}\), then \({L_{o} := \{ \Lambda [x]\,|\, x \in L \}}\) is a join-subcomplete sub-semilattice of L isomorphic to the order skeleton \({\tilde{L}}\) of L; for every order-preserving mapping f : L → Q from such a lattice L to a complete lattice Q, we define fo : Lo → Q by \({f_{o} := f|_{{L}_{o}}}\) and prove that \({u_{f} = u_{{f}_{o}}}\). For a lattice L with no infinite chains, the order skeleton \({\tilde{L}}\) of L is distributive if and only if the shadow σf of f is residuated for every complete lattice Q and every mapping f : L → Q. Related topics are discussed.

The β2-adrenoceptor agonist formoterol stimulates mitochondrial biogenesis.

Mitochondrial dysfunction is a common mediator of disease and organ injury. Although recent studies show that inducing mitochondrial biogenesis (MB) stimulates cell repair and regeneration, only a limited number of chemicals are known to induce MB. To examine the impact of the β-adrenoceptor (β-AR) signaling pathway on MB, primary renal proximal tubule cells (RPTC) and adult feline cardiomyocytes were exposed for 24 h to multiple β-AR agonists: isoproterenol (nonselective β-AR agonist), (±)-(R*,R*)-[4-[2-[[2-(3-chlorophenyl)-2-hydroxyethyl]amino]propyl]phenoxy] acetic acid sodium hydrate (BRL 37344) (selective β(3)-AR agonist), and formoterol (selective β(2)-AR agonist). The Seahorse Biosciences (North Billerica, MA) extracellular flux analyzer was used to quantify carbonylcyanide p-trifluoromethoxyphenylhydrazone (FCCP)-uncoupled oxygen consumption rate (OCR), a marker of maximal electron transport chain activity. Isoproterenol and BRL 37244 did not alter mitochondrial respiration at any of the concentrations examined. Formoterol exposure resulted in increases in both FCCP-uncoupled OCR and mitochondrial DNA (mtDNA) copy number. The effect of formoterol on OCR in RPTC was inhibited by the β-AR antagonist propranolol and the β(2)-AR inverse agonist 3-(isopropylamino)-1-[(7-methyl-4-indanyl)oxy]butan-2-ol hydrochloride (ICI-118,551). Mice exposed to formoterol for 24 or 72 h exhibited increases in kidney and heart mtDNA copy number, peroxisome proliferator-activated receptor γ coactivator 1α, and multiple genes involved in the mitochondrial electron transport chain (F0 subunit 6 of transmembrane F-type ATP synthase, NADH dehydrogenase subunit 1, NADH dehydrogenase subunit 6, and NADH dehydrogenase [ubiquinone] 1β subcomplex subunit 8). Cheminformatic modeling, virtual chemical library screening, and experimental validation identified nisoxetine from the Sigma Library of Pharmacologically Active Compounds and two compounds from the ChemBridge DIVERSet that increased mitochondrial respiratory capacity. These data provide compelling evidence for the use and development of β(2)-AR ligands for therapeutic MB.

SHADOWABLE CHAIN TRANSITIVE SETS OF C1-GENERIC DIFFEOMORPHISMS

We prove that a locally maximal chain transitive set of a C1-generic diffeomorphism is hyperbolic if and only if it is shadowable.

Maximal chains of Closed Prime Ideals for Discontinuous Algebra Norms on &lt;i&gt;C&lt;/i&gt;(&lt;i&gt;K&lt;/i&gt;)

Let K be an infinite compact space, let C(K) be the algebra of continuous complex-valued functions of K, let F be a well-ordered chain of nonmaximal prime ideals of C(K), let I_F be the smallest element of F and let M_F be the unique maximal ideal of C(K) containing the elements of F. Assuming the continuum hypothesis, we show that if |C(K)/IF | is the continuum , and if there exists a sequence (G_n ) of subsets of F U {M_F } stable under unions such that F U {MF } = UG_n , then there exists a discontinuous algebra norm p on C(K) such that the set of all nonmaximal prime ideals of C(K) which are closed with respect to p equals F.