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  • Artikel  (7)
  • 06A10  (7)
  • 2015-2019
  • 1980-1984  (7)
  • 1970-1974
  • 1950-1954
  • 1940-1944
  • 1984  (7)
  • Mathematik  (7)
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  • Artikel  (7)
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  • 2015-2019
  • 1980-1984  (7)
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  • Mathematik  (7)
  • 1
    Digitale Medien
    Digitale Medien
    Maximum antichains in the product of chains (1984)
    Springer
    Order 1 (1984), S. 21-28 
    Details
    ISSN: 1572-9273
    Schlagwort(e): 06A10 ; 05A05 ; Partially ordered sets ; Sperner's Theorem ; LYM property ; product of chains
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Let P be the poset k 1 × ... × k n , which is a product of chains, where n≥1 and k 1≥ ... ≥k n ≥2. Let $$M = k_1 - \sum\nolimits_{i = 2}^n {(k_i - 1)} $$ . P is known to have the Sperner property, which means that its maximum ranks are maximum antichains. Here we prove that its maximum ranks are its only maximum antichains if and only if either n=1 or M≤1. This is a generalization of a classical result, Sperner's Theorem, which is the case k 1= ... =k n =2. We also determine the number and location of the maximum ranks of P.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00396270
    Standort Signatur Erwartet Verfügbarkeit
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  • 2
    Digitale Medien
    Digitale Medien
    Finite cutsets and finite antichains (1984)
    Springer
    Order 1 (1984), S. 35-46 
    Details
    ISSN: 1572-9273
    Schlagwort(e): 06A10 ; Ordered set ; chain ; antichains ; width ; cutset
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract An ordered set (P,≤) has the m cutset property if for each x there is a set Fx with cardinality less than m, such that each element of Fx is incomparable to x and {x} ∪ Fx meets every maximal chain of (P,≤). Let n be least, such that each element x of any P having the m cutset property belongs to some maximal antichain of cardinality less than n. We specify n for m 〈 w. Indeed, n-1=m= width P for m=1,2,n=5 if m=3 and n⩾ℵ1 if m ≥4. With the added hypothesis that every bounded chain has a supremum and infimum in P, it is shown that for 4⩽m⩽ℵ0, n=ℵ0. That is, if each element x has a finite cutset Fx, each element belongs to a finite maximal antichain.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00396272
    Standort Signatur Erwartet Verfügbarkeit
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  • 3
    Digitale Medien
    Digitale Medien
    Minimizing the jump number for partially ordered sets: A graph-theoretic approach (1984)
    Springer
    Order 1 (1984), S. 7-19 
    Details
    ISSN: 1572-9273
    Schlagwort(e): 06A10 ; 05C20 ; 68C25 ; Partial order ; linear extension ; jump number ; line digraph ; cyclomatic number ; spanning branching ; Eulerian digraph
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The purpose of this paper is to present a graph-theoretic approach to the jump number problem for N-free posets which is based on the observation that the Hasse diagram of an N-free poset is a line digraph. Therefore, to every N-free poset P we can assign another digraph which is the root digraph of the Hasse diagram of P. Using this representation we show that the jump number of an N-free poset is equal to the cyclomatic number of its root digraph and can be found (without producing any linear extension) by an algorithm which tests if a given poset is N-free. Moreover, we demonstrate that there exists a correspondence between optimal linear extensions of an N-free poset and spanning branchings of its root digraph. We provide also another proof of the fact that optimal linear extensions of N-free posets are exactly greedy linear extensions. In conclusion, we discuss some possible generalizations of these results to arbitrary posets.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00396269
    Standort Signatur Erwartet Verfügbarkeit
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  • 4
    Digitale Medien
    Digitale Medien
    Powers of ordered sets (1984)
    Springer
    Order 1 (1984), S. 83-92 
    Details
    ISSN: 1572-9273
    Schlagwort(e): 06A10 ; Ordered sets ; exponentiation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Recently there has been significant progress in the study of powers of ordered sets. Much of this work has concerned cancellation laws for powers and uses these two steps. First, logarithmic operators are introduced to transform cancellation problems for powers into questions involving direct product decompositions. Second, refinement theorems for direct product decompositions are brought to bear. Here we present two results with the aim of highlighting these steps.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00396275
    Standort Signatur Erwartet Verfügbarkeit
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    Artikel (Nationallizenzen)
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  • 5
    Digitale Medien
    Digitale Medien
    Balancing poset extensions (1984)
    Springer
    Order 1 (1984), S. 113-126 
    Details
    ISSN: 1572-9273
    Schlagwort(e): 06A10 ; 68E05 ; Sorting ; comparison ; information theoretic bound ; linear extension
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We show that any finite partially ordered setP (not a total order) contains a pair of elementsx andy such that the proportion of linear extensions ofP in whichx lies belowy is between 3/11 and 8/11. A consequence is that the information theoretic lower bound for sorting under partial information is tight up to a multiplicative constant. Precisely: ifX is a totally ordered set about which we are given some partial information, and ife(X) is the number of total orderings ofX compatible with this information, then it is possible to sortX using no more thanC log2 e (X) comparisons whereC is approximately 2.17.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00565647
    Standort Signatur Erwartet Verfügbarkeit
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  • 6
    Digitale Medien
    Digitale Medien
    Ordered sets with no chains of ideals of a given type (1984)
    Springer
    Order 1 (1984), S. 159-172 
    Details
    ISSN: 1572-9273
    Schlagwort(e): 06A05 ; 06A10 ; Ordered sets ; ideals ; better quasi-order
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We study the possible order types of chains of ideals in an ordered set. Our main result is this. Given an indecomposable countable order type α, there is a finite listA 1 α , ...,A n α of ordered sets such that for every ordered setP the setJ(P) of ideals ofP, ordered by inclusion, contains a chain of type α if and only ifP contains a subset isomorphic to one of theA 1 #x03B1; , ...,A n α . The finiteness of the list relies on the notion of better quasi-ordering introduced by Nash-Williams and the properties of scattered chains obtained by Laver.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00565651
    Standort Signatur Erwartet Verfügbarkeit
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  • 7
    Digitale Medien
    Digitale Medien
    Compactness and subsets of ordered sets that meet all maximal chains (1984)
    Springer
    Order 1 (1984), S. 147-157 
    Details
    ISSN: 1572-9273
    Schlagwort(e): 06A10 ; 54D30 ; Maximal chain ; cutset ; chain complete ; antichain ; cofinal subset
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract LetP be a chain complete ordered set. By considering subsets which meet all maximal chains, we describe conditions which imply that the space of maximal chains ofP is compact. The symbolsP 1 andP 2 refer to two particular ordered sets considered below. It is shown that the space of maximal chains ℳ (P) is compact ifP satisfies any of the following conditions: (i)P contains no copy ofP 1 or its dual and all antichains inP are finite. (ii)P contains no properN and every element ofP belongs to a finite maximal antichain ofP. (iii)P contains no copy ofP 1 orP 2 and for everyx inP there is a finite subset ofP which is coinitial abovex. We also describe an example of an ordered set which is complete and densely ordered and in which no antichain meets every maximal chain.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00565650
    Standort Signatur Erwartet Verfügbarkeit
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