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  • Articles  (42)
  • 06A06  (42)
  • 1990-1994  (42)
  • 1965-1969
  • Mathematics  (42)
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  • Articles  (42)
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  • Mathematics  (42)
  • 1
    Electronic Resource
    Electronic Resource
    Every triangle-free planar graph has a planar upward drawing (1993)
    Springer
    Order 10 (1993), S. 1-16 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; Planar graph ; triangle-free ; upward drawing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A planar ordered set has a triangle-free, planar covering graph; on the other hand, there are nonplanar ordered sets whose covering graphs are planar. We show thatevery triangle-free planar graph has a planar upward drawing. This planar upward drawing can be constructed in time, polynomial in the number of vertices. Our results shed light on the apparently difficult problem, of long-standing, whether there is aneffective planarity-testing procedure for an ordered set.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01108704
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  • 2
    Electronic Resource
    Electronic Resource
    On the dimension of ordered sets with the 2-cutset property (1993)
    Springer
    Order 10 (1993), S. 37-54 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; Ordered set ; cutset ; 2-cutset property ; chain ; antichain ; width ; linear extension ; dimension
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An ordered setP is said to have 2-cutset property if, for every elementx ofP, there is a setS of elements ofP which are noncomparable tox, with |S|⩽2, such that every maximal chain inP meets {x}∪S. We consider the following question: Does there exist ordered sets with the 2-cutset property which have arbitrarily large dimension? We answer the question in the negative by establishing the following two results.Theorem: There are positive integersc andd such that every ordered setP with the 2-cutset property can be represented asP=X∪Y, whereX is an ordinal sum of intervals ofP having dimension ⩽d, andY is a subset ofP having width ⩽c. Corollary: There is a positive integern such that every ordered set with the 2-cutset property has dimension ⩽n.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01108707
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  • 3
    Electronic Resource
    Electronic Resource
    A note on sphere containment orders (1993)
    Springer
    Order 10 (1993), S. 143-145 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; Sphere containment order ; circle containment order
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Using the ideas of Scheinerman and Wierman [1] and of Hurlbert [2] we give a very short proof that the infinite order [2]×[3]×ω cannot be represented by containment of Euclidean balls in ad-dimensional space for anyd. Also we give representations of the orders [2]×[2]×ω and [3]×[3]×[3] by containment of circles in the plane.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01111298
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  • 4
    Electronic Resource
    Electronic Resource
    The core of a chain complete poset with no one-way infinite fence and no tower (1993)
    Springer
    Order 10 (1993), S. 349-361 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; Chain complete poset ; fixed point property ; perfect sequence ; core
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Like dismantling for finite posets, a perfect sequence Π = 〈P ξ :ξ≤λ〉 of a chain complete posetP represents a canonical procedure to produce a coreP λ. It has been proved that if the posetP contains no infinite antichain then this coreP λ is a retract ofP andP has the fixed point property iffP λ has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01108829
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  • 5
    Electronic Resource
    Electronic Resource
    Treewidth of cocomparability graphs and a new order-theoretic parameter (1994)
    Springer
    Order 11 (1994), S. 47-60 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; Treewidth ; pathwidth ; dimension ; interval dimension ; cocomparability graphs ; approximation algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We show that the pathwidth of a cocomparability graphG equals its treewidth. The proof is based on a new notion, calledinterval width, for a partial orderP, which is the smallest width of an interval order contained inP, and which is shown to be equal to the treewidth of its cocomparability graph. We observe also that determining any of these parameters isNP-hard and we establish some connections between classical dimension notions of partial orders and interval width. In fact we develop approximation algorithms for interval width ofP whose performance ratios depend on the dimension and interval dimension ofP, respectively.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01462229
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  • 6
    Electronic Resource
    Electronic Resource
    The strong fixed point property for small sets (1991)
    Springer
    Order 8 (1991), S. 29-32 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; (strong) fixed point property
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is shown that every partially ordered set with the fixed point property and with ten or fewer elements actually has the strong fixed point property.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00385811
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  • 7
    Electronic Resource
    Electronic Resource
    The dimension of orthomodular posets constructed by pasting Boolean algebras (1993)
    Springer
    Order 10 (1993), S. 183-197 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; 06A07 ; Dimension ; Boolean algebra ; atomic amalgam
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The main aim of this paper is the calculation of the dimension of certain atomic amalgams. These consist of finite Boolean algebras (blocks) pasted together in such a way that a pair of blocks intersects either trivially in the bounds, or the intersection consists of the bounds, an atom, and its complement.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01111302
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  • 8
    Electronic Resource
    Electronic Resource
    Spherical containment and the Minkowski dimension of partial orders (1993)
    Springer
    Order 10 (1993), S. 227-237 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; 52A37 ; Causal order ; spherical (containment) order ; Minkowski dimension
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The recent work on circle orders generalizes to higher dimensional spheres. As spherical containment is equivalent to causal precedence in Minkowski space, we define the Minkowski dimension of a poset to be the dimension of the minimal Minkowski space into which the poset can be embedded; this isd if the poset can be represented by containment with spheresS d−2 and of no lower dimension. Comparing this dimension with the standard dimension of partial orders we prove that they are identical in dimension two but not in higher dimensions, while their irreducible configurations are the same in dimensions two and three. Moreover, we show that there are irreducible configurations for arbitrarily large Minkowski dimension, thus providing a lower bound for the Minkowski dimension of partial orders.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01110544
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  • 9
    Electronic Resource
    Electronic Resource
    Transforming binary sequences using priority queues (1993)
    Springer
    Order 10 (1993), S. 31-36 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; Priority queue ; binary sequence ; enumeration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A priority queue transforms an input sequence σ into an output sequence τ which is a re-ordering of the sequence σ. The setR of all such related pairs is studied in the case that σ is a binary sequence. It is proved thatR is a partial order and that ¦R¦=c n+1, the (n+1)th Catalan number. An efficient (O(n 2)) algorithm is given for computing the number of outputs achievable from a given input.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01108706
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  • 10
    Electronic Resource
    Electronic Resource
    A chain complete poset with no infinite antichain has a finite core (1993)
    Springer
    Order 10 (1993), S. 55-63 
    Details
    ISSN: 1572-9273
    Keywords: 06A06 ; Poset ; PT-order ; chain complete ; retract ; fixed-point ; core
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is well known that dismantling a finite posetP leads to a retract, called the core ofP, which has the fixed-point property if and only ifP itself has this property. The PT-order, or passing through order, of a posetP is the quasi order ⊴ defined onP so thata⊴b holds if and only if every maximal chain ofP which passes througha also passes throughb. This leads to a generalization of the dismantling procedure which works for arbitrary chain complete posets which have no infinite antichain. We prove that such a poset also has a finite core, i.e. a finite retract which reflects the fixed-point property forP.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01108708
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