scholarly journals Maximal antichains of isomorphic subgraphs of the Rado graph

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1919-1923 ◽  
Author(s):  
Milos Kurilic ◽  
Petar Markovic
Keyword(s):  
Positive Integer ◽  
Partial Order ◽  
Maximal Antichain ◽  
Chain Complete ◽  
A Chain

If ?R,E? is the Rado graph andR(R) the set of its copies inside R, then ?R(R), ?? is a chain-complete and non-atomic partial order of the size 2x0 . A family A ? R(R) is a maximal antichain in this partial order iff (1) A ? B does not contain a copy of R, for each different A, B ?A and (2) For each S ? R(R) there is A ? A such that A ? S contains a copy of R. We show that the partial order ?R(R), ?? contains maximal antichains of size 2x0, X0 and n, for each positive integer n (thus, of all possible cardinalities, under CH). The results are compared with the corresponding known results concerning the partial order ?[?]?, ??.

scholarly journals A note on local polynomial functions on commutative semigroups

1982 ◽  
Vol 33 (1) ◽  
pp. 50-53
Author(s):  
P. A. Grossman
Keyword(s):  
Positive Integer ◽  
Universal Algebra ◽  
Polynomial Function ◽  
Polynomial Functions ◽  
Commutative Semigroups ◽  
Local Polynomial ◽  
A Chain

AbstractGiven a universal algebra A, one can define for each positive integer n the set of functions on A which can be “interpolated” at any n elements of A by a polynomial function on A. These sets form a chain with respect to inclusion. It is known for several varieties that many of these sets coincide for all algebras A in the variety. We show here that, in contrast with these results, the coincident sets in the chain can to a large extent be specified arbitrarily by suitably choosing A from the variety of commutative semigroups.

scholarly journals CATEGORIFICATION OF THE DICHROMATIC POLYNOMIAL FOR GRAPHS

2008 ◽  
Vol 17 (01) ◽  
pp. 31-45 ◽  
Author(s):  
MARKO STOŠIĆ
Keyword(s):  
Positive Integer ◽  
Euler Characteristic ◽  
Jones Polynomial ◽  
Chain Complex ◽  
Tutte Polynomial ◽  
A Chain ◽  
Alternating Link ◽  
Dichromatic Polynomial

For each graph and each positive integer n, we define a chain complex whose graded Euler characteristic is equal to an appropriate n-specialization of the dichromatic polynomial. This also gives a categorification of n-specializations of the Tutte polynomial of graphs. Also, for each graph and integer n ≤ 2, we define the different one-variable n-specializations of the dichromatic polynomial, and for each polynomial, we define graded chain complex whose graded Euler characteristic is equal to that polynomial. Furthermore, we explicitly categorify the specialization of the Tutte polynomial for graphs which corresponds to the Jones polynomial of the appropriate alternating link.

A chain complete poset with no infinite antichain has a finite core

Order ◽  
1993 ◽  
Vol 10 (1) ◽  
pp. 55-63 ◽  
Author(s):  
Boyu Li ◽  
E. C. Milner
Keyword(s):  
Chain Complete ◽  
A Chain

The Gap Between Partial and Full

1998 ◽  
Vol 08 (03) ◽  
pp. 399-430 ◽  
Author(s):  
J. Almeida ◽  
M. V. Volkov
Keyword(s):  
Partial Order ◽  
Finite Chain ◽  
Real Numbers ◽  
A Chain ◽  
Usual Order

We show that the interval of the lattice of semigroup pseudovarieties between the pseudovarieties generated by all semigroups of full and respectively, partial, order-preserving transformations of a finite chain, contains a chain isomorphic to the chain of real numbers (with the usual order). Similar results are proved for several related intervals.

scholarly journals The Möbius function of generalized subword order

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Peter R. W. McNamara ◽  
Bruce E. Sagan
Keyword(s):  
Partial Order ◽  
Finite Length ◽  
Simple Formula ◽  
Möbius Function ◽  
Finite Poset ◽  
Mobius Function ◽  
Nous Permet ◽  
International Audience ◽  
Uniform Manner ◽  
A Chain

International audience Let $P$ be a poset and let $P^*$ be the set of all finite length words over $P$. Generalized subword order is the partial order on $P^*$ obtained by letting $u≤ w$ if and only if there is a subword $u'$ of $w$ having the same length as $u$ such that each element of $u$ is less than or equal to the corresponding element of $u'$ in the partial order on $P$. Classical subword order arises when $P$ is an antichain, while letting $P$ be a chain gives an order on compositions. For any finite poset $P$, we give a simple formula for the Möbius function of $P^*$ in terms of the Möbius function of $P$. This permits us to rederive in an easy and uniform manner previous results of Björner, Sagan and Vatter, and Tomie. We are also able to determine the homotopy type of all intervals in $P^*$ for any finite $P$ of rank at most 1. Soit $P$ un ensemble partiellement ordonné et soit $P^*$ l'ensemble des mots de longueur finie sur $P$. On définit l'ordre des sous-mots généralisé comme l'ordre partiel sur $P^*$ obtenu en posant $u≤ w$ s'il existe un sous-mot $u'$ de $w$ ayant la même longueur que $u$, tel que chaque élément de $u$ soit plus petit ou égal à l'élément correspondant de $u'$ dans l'ordre partiel sur $P$. L'ordre des sous-mots classique correspond au cas où $P$ est une antichaîne ; tandis que si P est une chaîne, on obtient un ordre sur les compositions. Pour tout ensemble partiellement ordonné fini $P$, nous donnons une formule simple pour la fonction de Möbius de $P^*$ en fonction de celle de $P$. Cela nous permet de retrouver de manière simple et uniforme des résultats de Björner, Sagan et Vatter, et de Tomie. Nous sommes aussi en mesure de déterminer le type d'homotopie de tous les intervalles de $P^*$ pour n'importe quel $P$ fini de rang au plus 1.

$\mathcal{A}_t$-GROUPS SATISFYING A CHAIN CONDITION

2014 ◽  
Vol 13 (04) ◽  
pp. 1350137 ◽  
Author(s):  
LIHUA ZHANG ◽  
HAIPENG QU
Keyword(s):  
Positive Integer ◽  
Chain Condition ◽  
Metacyclic Group ◽  
Abelian Subgroups ◽  
A Chain

For a positive integer t, a finite p-group G is an [Formula: see text]-group if all subgroups of index pt in G are abelian, and at least one subgroup of index pt-1 in G is not abelian. An [Formula: see text]-group G satisfies a chain condition if every [Formula: see text]-subgroup of G is contained in an [Formula: see text]-subgroup for all i ∈ {0, 1, 2, …, t - 1}, where [Formula: see text]-subgroups denote abelian subgroups. We prove that if t ≥ 2, then, except for some p-groups of small order, the following conditions for a group G of order pn are equivalent: (1) G is an [Formula: see text]-group satisfying a chain condition; (2) every subgroup of order pn-k in G is an [Formula: see text]-subgroup for 0 ≤ k ≤ t; (3) G is an ordinary metacyclic group.

A note on the ascending chain condition of ideals

2019 ◽  
Vol 19 (07) ◽  
pp. 2050135 ◽  
Author(s):  
Ibrahim Al-Ayyoub ◽  
Malik Jaradat ◽  
Khaldoun Al-Zoubi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Chain Condition ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Regular Ideal ◽  
Reduction Number ◽  
Number Formula ◽  
A Chain ◽  
Ascending Chain Condition

We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that reach arbitrary long sequences of equalities, however the chain does not become stationary at that point. For a regular ideal [Formula: see text] in [Formula: see text], the Ratliff–Rush reduction number [Formula: see text] of [Formula: see text] is the smallest positive integer [Formula: see text] at which the chain [Formula: see text] becomes stationary. We construct ideals [Formula: see text] so that such a chain reaches an arbitrary long sequence of equalities but [Formula: see text] is not being reached yet.

scholarly journals Chain-finite operators and locally chain-finite operators

2001 ◽  
Vol 43 (1) ◽  
pp. 113-121
Author(s):  
Teresa Bermúdez ◽  
Antonio Martinón
Keyword(s):  
Positive Integer ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Generalized Inverse ◽  
Algebraic Characterization ◽  
Bounded Linear ◽  
A Chain

We give algebraic conditions characterizing chain-finite operators and locally chain-finite operators on Banach spaces. For instance, it is shown that T is a chain-finite operator if and only if some power of T is relatively regular and commutes with some generalized inverse; that is there are a bounded linear operator B and a positive integer k such that TkBTk =Tk and TkB=BTk. Moreover, we obtain an algebraic characterization of locally chain-finite operators similar to (1).

scholarly journals On the existence of sequences of co-prime pairs of integers

1989 ◽  
Vol 47 (1) ◽  
pp. 84-89 ◽  
Author(s):  
David L. Dowe
Keyword(s):  
Positive Integer ◽  
Supporting Evidence ◽  
Property A ◽  
A Chain ◽  
Positive Integers

AbstractWe say that a positive integer d has property (A) if for all positive integers m there is an integer x, depending on m, such that, setting n = m + d, x lies between m and n and x is co-prime to mn. We show that infinitely many even d and infinitely many odd d have property (A) and that infinitely many even d do not have property (A). We conjecture and provide supporting evidence that all odd d have property (A).Following A. R. Woods [3] we then describe conditions (Au) (for each u) asserting, for a given d, the existence of a chain of at most u + 2 integers, each co-prime to its neighbours, which start with m and increase, finishing at n = m + d. Property (A) is equivalent to condition (A1), and it is easily shown that property (Ai) implies property (Ai+1). Woods showed that for some u all d have property (Au), and we conjecture and provide supporting evidence that the least such u is 2.

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