Maximal Chains Research Articles

Maximal chains in the Turing degrees

AbstractWe study the problem of existence of maximal chains in the Turing degrees. We show that:1. ZF + DC + “There exists no maximal chain in the Turing degrees” is equiconsistent with ZFC + “There exists an inaccessible cardinal”2. For all a ∈ 2ω, (ω1)L[a] = ω1 if and only if there exists a [a] maximal chain in the Turing degrees. As a corollary, ZFC + “There exists an inaccessible cardinal” is equiconsistent with ZFC + “There is no (bold face) maximal chain of Turing degrees”.

A condition for modular lattices

This paper proves that a geometric lattice of rank n is a modular lattice if its every maximal chain contains a modular element of rank greater than 1 and less than n. This result is generalized to a more general lattices of finite rank.

The Maximal Chains of the Extended Bruhat Orders on the 𝒲 × 𝒲-Orbits of an Infinite Renner Monoid

Let (𝒲, S) be a Coxeter system. For ε, δ ∈ {+, −} we introduce and investigate combinatorially certain partial orders ≤ εδ, called extended Bruhat orders, on a 𝒲 × 𝒲-set 𝒲(N, C), which depends on 𝒲, a subset N ⊆ S, and a component C ⊆ N. We determine the length of the maximal chains between two elements x, y ∈ 𝒲(N, C), x ≤ εδ y. These posets generalize 𝒲 equipped with its Bruhat order. They include the 𝒲 × 𝒲-orbits of the Renner monoids of reductive algebraic monoids and of some infinite-dimensional generalizations which are equipped with the partial orders obtained by the closure relations of the Bruhat and Birkhoff cells. They also include the 𝒲 × 𝒲-orbits of certain posets obtained by generalizing the closure relation of the Bruhat cells of the wonderful compactification.

Comparing bacterial genomes from linear orders of patterns

We compare complete genomes from common words denoted MUMs for maximum unique matches. They allow to transform each genome into a linear order. We first evaluate the minimum length of a MUM shared by two genomes to be significant. Secondly, we compute maximal common chains of elements, that are in the same order in genomes. From these chains we define conserved genome segments as long DNA fragments having MUMs in the same order and with a bounded gap length between them. The resulting small number of segments allow to detect main evolutionary events as reversal or transposition of these fragments.

ON MAXIMAL CHAINS IN POSETS WITH GROUP ACTIONS

Our main purpose in this work is to study the maximal chains in group-posets to observe that this study gives us indications on the type of some group actions on posets. Therefore we shall study the behavior of the group actions on chains .

Properties of Auto- and Antiautomorphisms of Maximal Chain Structures and their Relations to i-Perspectivities

Let E be a non empty set, let P : = E × E, \({\mathfrak{G}_{1}}\) := {x × E|x ∈ E}, \({\mathfrak{G}_{2}}\) := {E × x|x ∈ E}, and \(\mathfrak{C}\) := {C ∈ 2P |∀X ∈ \(\mathfrak{G}_{1} \cup \mathfrak{G}_{2}\) : |C ∩ X| = 1} and let \(\mathfrak{B} \subseteq \mathfrak{C}\). Then the quadruple \((P, \mathfrak{G}_{1}, \mathfrak{G}_{2}, \mathfrak{B})\) resp. \((P, \mathfrak{G}_{1}, \mathfrak{G}_{2}, \mathfrak{C})\) is called chain structure resp. maximal chain structure. We consider the maximal chain structure \((P, \mathfrak{G}_{1}, \mathfrak{G}_{2}, \mathfrak{C})\) as an envelope of the chain structure \((P, \mathfrak{G}_{1}, \mathfrak{G}_{2}, \mathfrak{B})\). Particular chain structures are webs, 2-structures, (coordinatized) affine planes, hyperbola structures or Minkowski planes. Here we study in detail the groups of automorphisms \(Aut(P, \mathfrak{G}_{1}, \mathfrak{G}_{2})\), \(Aut(P, \mathfrak{G}_{1}, \cup \, \mathfrak{G}_{2})\), \(Aut(P, {\mathfrak{C}})\), \(Aut(P, {\mathfrak{G}_{1}}, {\mathfrak{G}_{2}}, {\mathfrak{C}})\) related to a maximal chain structure \((P, {\mathfrak{G}_{1}}, {\mathfrak{G}_{2}}, {\mathfrak{C}})\). The set \(\mathfrak C\) of all chains can be turned in a group \((\mathfrak {C}, ·)\) such that the subgroup \(\widehat{\mathfrak C}\) of \(Sym \mathfrak{C}\) generated by \({\mathfrak{C}_{l}}\) the left-, by \({\mathfrak{C}_{r}}\) the right-translations and by ι the inverse map of \((\mathfrak{C}, ·)\) is isomorphic to \(Aut(P, \mathfrak C)\) (cf. (2.14)).

Towers and maximal chains in Boolean algebras

A tower in a Boolean algebra A is a strictly increasing sequence, of regular length, of elements of A with supremum 1. We consider the following functions:

Cutset Condition for Geometric Lattices

In this paper we prove that an atomistic lattice L of finite length is geometric if it has the nontrivial modular cutset condition, that is, every maximal chain of L contains a modular element which is different from the minimum element and the maximum element of L.

Some Results on p-Nilpotence and Supersolvability of Finite Groups

Let G be a finite group. A subgroup K of a group G is called an ℋ-subgroup of G if N G (K) ∩ K x ≦ K for all x ∈ G. The set of all ℋ-subgroups of G will be denoted by ℋ(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P 0 ≨ P 1 ≨ ··· ≨ P n = P is called a maximal chain of P provided that |P i : P i−1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P 0 ≨ P 1 ≨ ··· ≨ P i ≨ ··· ≨ P n = P such that P i ∈ ℋ(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups.

The effect of nutrient availability and temperature on chain length of the diatom, Skeletonema costatum

We determined the effects of temperature and nutrients on the chain length of a diatom, Skeletonema costatum, in batch culture and enclosure experiments with estuarine water from San Francisco Bay, USA, using the recently developed CytoBuoy flow cytometer. Determination of the number of cells per diatom chain by CytoBuoy flow cytometer and associated software correlated well with but was much more precise and time efficient than microscopic quantification. Increasing temperatures (from 6, 8 to 17°C) and nutrient concentrations induced high growth rates and dominance by longer chains in a cultured S. costatum strain that was originally acclimatized to a temperature range of 11-30°C. Similarly, a positive correlation between growth rate and chain length was observed in S. costatum in batch culture and natural communities in enclosure experiments. Maximal chain lengths of S. costatum were greater in natural populations than in the batch culture. Longer chains affect sinking rates and thus likely help the diatom remain suspended in the upper part of the water column where physical and chemical parameters are more favorable for growth.

Comparison of Three Web Search Algorithms

In this paper we discuss three important kinds of Markov chains used in Web search algorithms-the maximal irreducible Markov chain, the minimal irreducible Markov chain and the middle irreducible Markov chain. We discuss the stationary distributions, the convergence rates and the Maclaurin series of the stationary distributions of the three kinds of Markov chains. Among other things, our results show that the maximal and minimal Markov chains have the same stationary distribution and that the stationary distribution of the middle Markov chain reflects the real Web structure more objectively. Our results also prove that the maximal and middle Markov chains have the same convergence rate and that the maximal Markov chain converges faster than the minimal Markov chain when the damping factor $$ \alpha > \frac{1} {{{\sqrt 2 }}} $$ .

Cutsets and anti-chains in linear lattices

Consider the poset, ordered by inclusion, of subspaces of a four-dimensional vector space over a field with 2 elements. We prove that, for this poset, any cutset (i.e., a collection of elements that intersects every maximal chain) contains a maximal anti-chain of the poset. In analogy with the same result by Duffus, Sands, and Winkler [D. Duffus, B. Sands, P. Winkler, Maximal chains and anti-chains in Boolean lattices, SIAM J. Discrete Math. 3 (2) (1990) 197–205] for the subset lattice, we conjecture that the above statement holds in any dimension and for any finite base field, and we prove some special cases to support the conjecture.

MORSE DECOMPOSITION, ATTRACTORS AND CHAIN RECURRENCE

The global behavior of a dynamical system can be described by its Morse decompositions or its attractor and repeller configurations. There is a close relation between these two approaches and also with (maximal) chain recurrent sets that describe the system behavior on finest Morse sets. These sets depend upper semicontinuously on parameters. The connection with ergodic theory is provided through the construction of invariant measures based on chains.

A numerical study of the correspondence between paths in a causal set and geodesics in the continuum

This paper presents the results of a computational study related to the path-geodesic correspondence in causal sets. For intervals in flat spacetimes, and in selected curved spacetimes, we present evidence that the longest maximal chains (the longest paths) in the corresponding causal set intervals statistically approach the geodesic for that interval in the appropriate continuum limit.

On generalized smooth groups

(Communicated by Ru¨diger Go¨bel)Abstract. A maximal chain in a finite lattice L is called smooth if any two intervals of the samelength are isomorphic. We call a lattice L totally smooth if all maximal chains of L are smooth.The main concept in this paper is that of a generalized smooth group, that is, a group G suchthat ½G=H is totally smooth for every subgroup H of G of prime order. The purpose of thispaper is to determine the structure of generalized smooth groups.2000 Mathematics Subject Classification: 20D30, 20E15.

Subgroups and supergroups of crystallographic and quasi-crystallographic point groups

The direct products and semidirect products of quasi-crystallographic point groups, were derived from group theory. According to crystallography theory, the stereographic projection of pentagonal system, octagonal system, decagonal system and dodecagonal system, were derived and drawn. Basing upon them, all maximal subgroups of each quasi-crystallographic point group were derived and drawn. As a result, subgroups and supergroups of the three-dimensional crystallographic and quasi-crystallographic point groups (the ‘family tree' of sixty point groups) were derived and depicted in drawing. In the ‘family tree', the minimal supergroups and maximal subgroups of sixty point groups were illustrated in detail in the form of maximal subgroup chains.

Fundamental semigroups for dynamical systems

Algebraic semigroups describing the dynamic behavior are associated to compact, locally maximal chain transitive subsets. The construction is based on perturbations and associated local control sets. The dependence on the perturbation structure is analyzed.

Graded left modular lattices are supersolvable

We provide a direct proof that a finite graded lattice with a maximal chain of left modular elements is supersolvable. This result was first established via a detour through EL-labellings in [MT] by combining results of McNamara [Mc] and Liu [Li]. As part of our proof, we show that the maximum graded quotient of the free product of a chain and a single-element lattice is finite and distributive.

On the ordering of benzenoid chains and cyclo-polyphenacenes with respect to their numbers of Clar aromatic sextets

Based on Clar aromatic sextet theory [Clar, The Aromatic Serxtet (Wiley, New York, 1972)] and the concept of sextet polynomial introduced by Hosoya and Yamaguchi [Mathematical Concepts in Organic Chemistry (Springer, Berlin, 1986)], we define a new ordering of benzenoid systems. For two isomeric benzenoid systems B1 and B2, we say B1>B2 if each coefficient of sextet polynomial of B1 is no less than the corresponding coefficient of sextet polynomial of B2. In this paper, we consider the ordering of the benzenoid chains. The maximal and second maximal benzenoid chains as well as the minimal, the second minimal up to the fourth minimal benzenoid chains are determined. Furthermore, under this ordering, we determine the maximal and second maximal cyclo-polyphenacenes as well as the minimal, the second minimal, and up to the seventh minimal cyclo-polyphenacenes.

On the FIP Property for Extensions of Commutative Rings #

ABSTRACT A (unital) extension R ⊆ T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ⊂ S ⊂ T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ⊂ T is a (module-) finite minimal ring extension, then R(X)⊂T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ⊆ T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ⊆ R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ⊆ R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ℤ which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (ℤ:R)≠0. Some results are also given for the residually FIP property.