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  • Life and Medical Sciences  (30,791)
  • 06A10
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  • 1
    Electronic Resource
    Electronic Resource
    Reflexive domains and fixed points (1985)
    Springer
    Acta applicandae mathematicae 4 (1985), S. 99-100 
    Details
    ISSN: 1572-9036
    Keywords: 03B99 ; 06A10 ; 54H25 ; Fixed points ; reflexive domains ; retracts ; posets
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In a recent paper in this journal, J. Soto-Andrade and F. J. Varela draw attention to the fact that ifR is a retract of a reflexive domain in a suitable category thenR has the fixed point property. They suggest [1], pp. 1 and 18, that conversely every structure with the fixed point property is a retract of a reflexive domain. In this note it is shown that ifR is a retract of a reflexive domain thenR R has the fixed point property. This leads to counterexamples to the suggestion of Soto-Andrade and Varela in the categoryPo of partially ordered sets and monotone maps.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02293493
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  • 2
    Electronic Resource
    Electronic Resource
    Random orders (1985)
    Springer
    Order 1 (1985), S. 317-331 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 60C05 ; Partially ordered set ; probabilistic methods ; dimension
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Letk andn be positive integers and fix a setS of cardinalityn; letP k (n) be the (partial) order onS given by the intersection ofk randomly and independently chosen linear orders onS. We begin study of the basic parameters ofP k (n) (e.g., height, width, number of extremal elements) for fixedk and largen. Our object is to illustrate some techniques for dealing with these ‘random orders’ and to lay the groundwork for future research, hoping that they will be found to have useful properties not obtainable by known constructions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00582738
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  • 3
    Electronic Resource
    Electronic Resource
    A companion to Grillet's Theorem on maximal chains and antichains (1985)
    Springer
    Order 1 (1985), S. 371-375 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; Poset ; chain ; antichain ; cutset
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is shown that ifP is a poset containing noN, then every minimal cutset inP is an antichain, that the converse also holds whenP is finite, and that this converse fails in general.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00582742
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  • 4
    Electronic Resource
    Electronic Resource
    Minimizing the jump number for partially ordered sets: A graph-theoretic approach (1984)
    Springer
    Order 1 (1984), S. 7-19 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 05C20 ; 68C25 ; Partial order ; linear extension ; jump number ; line digraph ; cyclomatic number ; spanning branching ; Eulerian digraph
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00396269
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  • 5
    Electronic Resource
    Electronic Resource
    Completions of orthomodular lattices II (1993)
    Springer
    Order 10 (1993), S. 283-294 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 20B14 ; Orthomodular lattices ; MacNeille completion ; Boolean algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract If $$\mathcal{K}$$ is a variety of orthomodular lattices generated by a set of orthomodular lattices having a finite uniform upper bound om the length of their chains, then the MacNeille completion of every algebra in $$\mathcal{K}$$ again belongs to $$\mathcal{K}$$ .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01110549
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  • 6
    Electronic Resource
    Electronic Resource
    Scheduling tasks with communication delays on parallel processors (1995)
    Springer
    Order 12 (1995), S. 213-220 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 68C25 ; Parallel computation ; m-machine problem ; tree
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetP={v 1,...,v n } be a set ofn jobs to be executed on a set ofm identical machines. In many instances of scheduling problems, if a jobv i has to be executed before the jobv j and both jobs are to be executed on different machines, some sort of information exchange has to take place between the machines executing them. The time it takes for this exchange of information is called a communication delay. In this paper we give anO(n) algorithm to find an optimal scheduling with communication delays when the number of machines is not limited and the precedence constraints on the jobs form a tree.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01111739
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  • 7
    Electronic Resource
    Electronic Resource
    Jump number problem: The role of matroids (1985)
    Springer
    Order 2 (1985), S. 1-8 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; Poset ; N-free poset ; linear extension ; Jump number ; matroid ; greedy algorithm ; Rado-Edmonds theorem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let L=x 1 x 2...x n be a linear extension of a poset P. Each pair (x i , x i+1) such that x i ≮ x i+1in P is called a jump of L. It is well known that for N-free posets a natural ‘greedy’ procedure constructing linear extensions yields a linear extension with a minimum number of jumps. We show that there is a matroid corresponding to any N-free poset and apply the Rado-Edmonds Theorem to obtain another proof of this result.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00337919
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  • 8
    Electronic Resource
    Electronic Resource
    Arithmetic progressions in partially ordered sets (1987)
    Springer
    Order 4 (1987), S. 37-42 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; Partially ordered sets ; Van der Waerden's arithmetic sequence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Van der Waerden's arithmetic sequence theorem—in particular, the ‘density version’ of Szemerédi—is generalized to partially ordered sets in the following manner. Let w and t be fixed positive integers and ε〉0. Then for every sufficiently large partially ordered set P of width at most w, every subset S of P satisfying |S|≥ε|P| contains a chain a 1, a 2,..., a 1 such that the cardinality of the interval [a i, a i+1] in P is the same for each i.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00346651
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  • 9
    Electronic Resource
    Electronic Resource
    NP-completeness properties about linear extensions (1987)
    Springer
    Order 4 (1987), S. 143-154 
    Details
    ISSN: 1572-9273
    Keywords: 06A10 ; 68C25 ; Jump number ; greedy linear extensions ; Dilworth posets ; satisfiability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Following the pioneering work of Kierstead, we present here some complexity results about the construction of depth-first greedy linear extensions. We prove that the recognition of Dilworth partially ordered sets of height 2, as defined by Syslo, is NP-complete. This last result yields a new proof of the NP-completeness of the jump number problem, first proved by Pulleyblank.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00337693
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  • 10
    Electronic Resource
    Electronic Resource
    Rational preemptive scheduling (1987)
    Springer
    Order 4 (1987), S. 195-206 
    Details
    ISSN: 1572-9273
    Keywords: 68C15 ; 06A10 ; Posets ; preemptive ; scheduling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In scheduling jobs subject to precedence constraints that form a partial order, it is advantageous to interrupt (preempt) jobs, and return to complete them at a later time in order to minimize total completion time. It is clearly desirable to see that such preemptive scheduling by finitely many machines requires only intervals of work, and not a more general assignment of tasks over measurable sets, for optimal completion. It is a deeper fact that arbitrarily small intervals are required for a fixed number of machines m〉-3 for optimal preemptive scheduling. On the other hand, if the number of jobs is fixed, say n, then it is intuitively clear that it suffices to use only comparatively large intervals (but less clear ‘how large’ will suffice!). The authors address these and certain related questions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00337697
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