scholarly journals Traces Without Maximal Chains

10.37236/465 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ta Sheng Tan
Keyword(s):  
Maximal Chain ◽  
Maximal Chains

The trace of a family of sets ${\cal A}$ on a set $X$ is ${\cal A}|_X=\{A\cap X:A\in {\cal A}\}$. If ${\cal A}$ is a family of $k$-sets from an $n$-set such that for any $r$-subset $X$ the trace ${\cal A}|_X$ does not contain a maximal chain, then how large can ${\cal A}$ be? Patkós conjectured that, for $n$ sufficiently large, the size of ${\cal A}$ is at most ${n-k+r-1\choose r-1}$. Our aim in this paper is to prove this conjecture.

Connectedness Properties of Lattices

1986 ◽  
Vol 29 (3) ◽  
pp. 314-320 ◽  
Author(s):  
R. Vainio
Keyword(s):  
Maximal Chain ◽  
Connected Components ◽  
Continuous Lattice ◽  
Order Convergence ◽  
Convergence Space ◽  
Chain Conditions ◽  
Interval Topology ◽  
Convergence Structure ◽  
Maximal Chains

AbstractLet L be a lattice and q a convergence structure (or a topology) finer than the interval topology of L. In case of compact maximal chains and continuous lattice translations, the connected components of the space (L,q) are characterized using lattice conditions only. Moreover, lattice conditions of L are related to connectedness conditions of the order convergence space (L, o). Throughout this note, maximal chain conditions and maximal chain techniques play an important role.

Maximal chains in the Turing degrees

2007 ◽  
Vol 72 (4) ◽  
pp. 1219-1227 ◽  
Author(s):  
C. T. Chong ◽  
Liang Yu
Keyword(s):  
Maximal Chain ◽  
Turing Degrees ◽  
Inaccessible Cardinal ◽  
Maximal Chains ◽  
Bold Face

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”.

scholarly journals Deformation of Chains via a Local Symmetric Group Action

10.37236/1459 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Patricia Hersh
Keyword(s):  
Symmetric Group ◽  
Group Action ◽  
Maximal Chain ◽  
Type B ◽  
Noncrossing Partition ◽  
Partition Lattices ◽  
Maximal Chains ◽  
Symmetric Chain

A symmetric group action on the maximal chains in a finite, ranked poset is local if the adjacent transpositions act in such a way that $(i,i+1)$ sends each maximal chain either to itself or to one differing only at rank $i$. We prove that when $S_n$ acts locally on a lattice, each orbit considered as a subposet is a product of chains. We also show that all posets with local actions induced by labellings known as $R^* S$-labellings have symmetric chain decompositions and provide $R^* S$-labellings for the type B and D noncrossing partition lattices, answering a question of Stanley.

Antichains and Finite Sets that Meet all Maximal Chains

1986 ◽  
Vol 38 (3) ◽  
pp. 619-632 ◽  
Author(s):  
J. Ginsburg ◽  
I. Rival ◽  
B. Sands
Keyword(s):  
Topological Space ◽  
Finite Subset ◽  
Ordered Set ◽  
Maximal Chain ◽  
Product Topology ◽  
Finite Sets ◽  
Power Set ◽  
Image Position ◽  
Maximal Chains ◽  
Usual Product

This paper is inspired by two apparently different ideas. Let P be an ordered set and let M(P) stand for the set of all of its maximal chains. The collection of all sets of the formandwhere x ∊ P, is a subbase for the open sets of a topology on M(P). (Actually, it is easy to check that the B(x) sets themselves form a subbase.) In other words, as M(P) is a subset of the power set 2|p| of P, we can regard M(P) as a subspace of 2|p| with the usual product topology. M. Bell and J. Ginsburg [1] have shown that the topological space M(P) is compact if and only if, for each x ∊ P, there is a finite subset C(x) of P all of whose elements are noncomparable to x and such that {x} ∪ C(x) meets each maximal chain.

scholarly journals For Which Finite Groups G is the Lattice ℒ(G) of Subgroups Gorenstein?

1987 ◽  
Vol 105 ◽  
pp. 147-151 ◽  
Author(s):  
Takayuki Hibi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Classical Result ◽  
Maximal Chain ◽  
Finite Lattice ◽  
Normal Subgroups ◽  
A Finite Group ◽  
Maximal Chains

Let G be a finite group and ℒ(G) the lattice consisting of all subgroups of G. It is well known that ℒ(G) is distributive if and only if G is cyclic (cf. [2, p. 173]). Moreover, the classical result of Iwasawa [8] says that ℒ(G) is pure if and only if G is supersolvable. Here, a finite lattice is called pure if all of maximal chains in it have same length and a finite group G is called supersolvable if ℒ(G) has a maximal chain which consists of normal subgroups of G.

Finite maximal chains of commutative rings

2014 ◽  
Vol 14 (01) ◽  
pp. 1450075 ◽  
Author(s):  
Ahmed Ayache ◽  
David E. Dobbs
Keyword(s):  
Maximal Chain ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Quotient Field ◽  
One Dimensional ◽  
Normal Pair ◽  
Intermediate Ring ◽  
Additional Domain ◽  
Maximal Chains ◽  
Dimensional Domain

Let R ⊆ S be a unital extension of commutative rings, with [Formula: see text] the integral closure of R in S, such that there exists a finite maximal chain of rings from R to S. Then S is a P-extension of R, [Formula: see text] is a normal pair, each intermediate ring of R ⊆ S has only finitely many prime ideals that lie over any given prime ideal of R, and there are only finitely many [Formula: see text]-subalgebras of S. Each chain of rings from R to S is finite if dim (R) = 0; or if R is a Noetherian (integral) domain and S is contained in the quotient field of R; or if R is a one-dimensional domain and S is contained in the quotient field of R; but not necessarily if dim (R) = 2 and S is contained in the quotient field of R. Additional domain-theoretic applications are given.

scholarly journals Promotion and Evacuation

10.37236/75 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Vector Space ◽  
Hecke Algebra ◽  
Linear Extensions ◽  
Linear Transformations ◽  
Basic Properties ◽  
Finite Poset ◽  
Order Ideals ◽  
Maximal Chains

Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schützenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.

scholarly journals Fixed Points of the Evacuation of Maximal Chains on Fuss Shapes

10.37236/6898 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Hsiang-Chun Hsu ◽  
Yu-Pei Huang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Rectangular Shape ◽  
Explicit Characterization ◽  
Finite Poset ◽  
Maximal Chains

For a partition $\lambda$ of an integer, we associate $\lambda$ with a slender poset $P$ the Hasse diagram of which resembles the Ferrers diagram of $\lambda$. Let $X$ be the set of maximal chains of $P$. We consider Stanley's involution $\epsilon:X\rightarrow X$, which is extended from Schützenberger's evacuation on linear extensions of a finite poset. We present an explicit characterization of the fixed points of the map $\epsilon:X\rightarrow X$ when $\lambda$ is a stretched staircase or a rectangular shape. Unexpectedly, the fixed points have a nice structure, i.e., a fixed point can be decomposed in half into two chains such that the first half and the second half are the evacuation of each other. As a consequence, we prove anew Stembridge's $q=-1$ phenomenon for the maximal chains of $P$ under the involution $\epsilon$ for the restricted shapes.

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