• Title, Summary, Keyword: finite ordered set

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SOME FAMILIES OF IDEAL-HOMOGENEOUS POSETS

  • Chae, Gab-Byung;Cheong, Minseok;Kim, Sang-Mok
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.971-983
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    • 2016
  • A partially ordered set P is ideal-homogeneous provided that for any ideals I and J, if $$I{\sim_=}_{\sigma}J$$, then there exists an automorphism ${\sigma}^*$ such that ${\sigma}^*{\mid}_I={\sigma}$. Behrendt [1] characterizes the ideal-homogeneous partially ordered sets of height 1. In this paper, we characterize the ideal-homogeneous partially ordered sets of height 2 and nd some families of ideal-homogeneous partially ordered sets.

ON THE DIMENSION OF AMALGAMATED ORDERED SETS

  • Lee, Jeh-Gwon
    • Bulletin of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.117-123
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    • 1992
  • The dimension problem has been one of central themes in the theory of ordered sets. In this paper we focus on amalgamated ordered sets. Although some results can be obviously applied to infinite cases, we assume throughout that all ordered set are finite. If A and B are ordered sets whose orders agree on A.cap.B, then the amalgam of A and B is defined to the the set A.cup.B in which the order is the transitive closure of the union of the two orders, i.e., the smallest order containing the two orders, and is denoted by A .or. B .leq. dim A + dim B for any ordered sets A and B. But it is quite surprising that the dimension of the amalgam of certain 2-dimensional ordered sets can be arbitrarily large.

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Inversion-like and Major-like Statistics of an Ordered Partition of a Multiset

  • Choi, Seung-Il
    • Kyungpook Mathematical Journal
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    • v.56 no.3
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    • pp.657-668
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    • 2016
  • Given a partition ${\lambda}=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_l)$ of a positive integer n, let Tab(${\lambda}$, k) be the set of all tabloids of shape ${\lambda}$ whose weights range over the set of all k-compositions of n and ${\mathcal{OP}}^k_{\lambda}_{rev}$ the set of all ordered partitions into k blocks of the multiset $\{1^{{\lambda}_l}2^{{\lambda}_{l-1}}{\cdots}l^{{\lambda}_1}\}$. In [2], Butler introduced an inversion-like statistic on Tab(${\lambda}$, k) to show that the rank-selected $M{\ddot{o}}bius$ invariant arising from the subgroup lattice of a finite abelian p-group of type ${\lambda}$ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(${\lambda}$, k) and ${\mathcal{OP}}^k_{\hat{\lambda}}$. When k = 2, we also introduce a major-like statistic on Tab(${\lambda}$, 2) and study its connection to the inversion statistic due to Butler.

DUAL ALGORITHM FOR $GL_1$ ISOTONIC OPTIMIZATION WITH WEIGHTS ON A PARTIALLY ORDERED SET

  • Chung, Seiyoung
    • Bulletin of the Korean Mathematical Society
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    • v.28 no.2
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    • pp.243-254
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    • 1991
  • For a given function f.mem.F and a set of functions J.subeq.F, the problem of isotonic optimization is to determine an element in the set nearest to f in some sense. Specifically, let X be a partially ordered finite set with a partial order << and, let F"=F(X) be the linear space of all bounded real valued functions on X. A function g.mem.F is said to be an isotonic function if g(x).leq.g(y) whenever x,y.mem.X and x << y.<< y.

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A Mixed Model for Oredered Response Categories

  • Choi, Jae-Sung
    • Journal of the Korean Data and Information Science Society
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    • v.15 no.2
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    • pp.339-345
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    • 2004
  • This paper deals with a mixed logit model for ordered polytomous data. There are two types of factors affecting the response varable in this paper. One is a fixed factor with finite quantitative levels and the other is a random factor coming from an experimental structure such as a randomized complete block design. It is discussed how to set up the model for analyzing ordered polytomous data and illustrated how to estimate the paramers in the given model.

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Min-Max Regret Version of an m-Machine Ordered Flow Shop with Uncertain Processing Times

  • Park, Myoung-Ju;Choi, Byung-Cheon
    • Management Science and Financial Engineering
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    • v.21 no.1
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    • pp.1-9
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    • 2015
  • We consider an m-machine flow shop scheduling problem to minimize the latest completion time, where processing times are uncertain. Processing time uncertainty is described through a finite set of processing time vectors. The objective is to minimize maximum deviation from optimality for all scenarios. Since this problem is known to be NP-hard, we consider it with an ordered property. We discuss optimality properties and develop a pseudo-polynomial time approach for the problem with a fixed number of machines and scenarios. Furthermore, we find two special structures for processing time uncertainty that keep the problem NP-hard, even for two machines and two scenarios. Finally, we investigate a special structure for uncertain processing times that makes the problem polynomially solvable.

A NOTE ON NULL DESIGNS OF DUAL POLAR SPACES

  • CHO, SOO-JIN
    • Communications of the Korean Mathematical Society
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    • v.20 no.1
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    • pp.15-21
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    • 2005
  • Null designs on the poset of dual polar spaces are considered. A poset of dual polar spaces is the set of isotropic subspaces of a finite vector space equipped with a nondegenerate bilinear form, ordered by inclusion. We show that the minimum number of isotropic subspaces to construct a nonzero null t-design is ${\prod}^{t}_{i=0}(1+q^{i})$ for the types $B_N,\;D_N$, whereas for the case of type $C_N$, more isotropic subspaces are needed.

Numerical measures of Indicating Placement of Posets on Scale from Chains to Antichains

  • Bae, Kyoung-Yul
    • The Journal of Information Technology and Database
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    • v.3 no.1
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    • pp.97-108
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    • 1996
  • In this paper we obtain several function defined on finite partially ordered sets(posets) which may indicate constraints of comparability on sets of teams(tasks, etc.) for which evaluation is computationally simple, a relatively rare condition in graph-based algorithms. Using these functions a set of numerical coefficients and associated distributions obtained from a computer simulation of certain families of random graphs is determined. From this information estimates may be made as to the actual linearity of complicated posets. Applications of these ideas is to all areas where obtaining rankings from partial information in rational ways is relevant as in, e.g., team_, scaling_, and scheduling theory as well as in theoretical computer science. Theoretical consideration of special and desirable properties of various functions is provided permitting judgment concerning sensitivity of these functions to changes in parameters describing (finite) posets.

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COREGULARITY OF ORDER-PRESERVING SELF-MAPPING SEMIGROUPS OF FENCES

  • JENDANA, KETSARIN;SRITHUS, RATANA
    • Communications of the Korean Mathematical Society
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    • v.30 no.4
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    • pp.349-361
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    • 2015
  • A fence is an ordered set that the order forms a path with alternating orientation. Let F = (F;${\leq}$) be a fence and let OT(F) be the semigroup of all order-preserving self-mappings of F. We prove that OT(F) is coregular if and only if ${\mid}F{\mid}{\leq}2$. We characterize all coregular elements in OT(F) when F is finite. For any subfence S of F, we show that the set COTS(F) of all order-preserving self-mappings in OT(F) having S as their range forms a coregular subsemigroup of OT(F). Under some conditions, we show that a union of COTS(F)'s forms a coregular subsemigroup of OT(F).