## Relative blocking in posets

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Venue: | J. Comb. Optim |

Citations: | 7 - 7 self |

### BibTeX

@ARTICLE{Matveev_relativeblocking,

author = {Andrey O. Matveev},

title = {Relative blocking in posets},

journal = {J. Comb. Optim},

year = {},

pages = {379--403}

}

### OpenURL

### Abstract

Abstract. Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences. 1. Introduction and

### Citations

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Citation Context ...to be the ascending sequence of all irreducible fractions between 0 and 1 whose denominators do not exceed n, see, e.g., Chapter 27 of Buchstab, 1967, Chapter 4 of Graham et al., 1994, Chapter III of =-=Hardy and Wright, 1979-=-, and Lagarias and Tresser, 1995. Thus, F(P, a; ω) is a subsequence of the Farey sequence of order ω(P). We always index the fractions from F(P, a; ω) starting with zero: F(P, a; ω) = ( f0 := 0 1 < f1... |

150 |
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Citation Context ...ts order ideal I(A) is assigned the isomorphic face poset of the abstract simplicial complex whose facets are the sets from the family {I(a) ∩ B(n) (1) : a ∈ A}. See, e.g., Billera and Björner, 1997, =-=Björner, 1995-=-, Bruns and Herzog,8 ANDREY O. MATVEEV 1998, Buchstaber and Panov, 2004, Hibi, 1992, Miller and Sturmfels, 2004, Stanley, 1996, and Ziegler, 1998, on simplicial complexes. The following proposition l... |

139 |
Combinatorics and commutative algebra, second edition
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Citation Context ... family {I(a) ∩ B(n) (1) : a ∈ A}. See, e.g., Billera and Björner, 1997, Björner, 1995, Bruns and Herzog,8 ANDREY O. MATVEEV 1998, Buchstaber and Panov, 2004, Hibi, 1992, Miller and Sturmfels, 2004, =-=Stanley, 1996-=-, and Ziegler, 1998, on simplicial complexes. The following proposition lists some observations. Proposition 2.3. holds (i) If A is a nontrivial antichain in P, then it Ir(P, A; ω) = ⋂ Ir(P, a; ω) , a... |

139 | Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes. Memoirs - Zaslavsky - 1975 |

127 |
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(Show Context)
Citation Context ...mittees, undertaken in the paper, has been partly motivated by the need for a more detailed analysis of building blocks of decision rules in applied contradictory problems of pattern recognition. See =-=Duda et al., 2001-=-, on the setting of the pattern recognition problem and various methods to solve it. Consider a finite nonempty collection H := {H1, . . ., Hm} of codimension one linear subspaces Hi := {x ∈ Rn : 〈pi,... |

90 | The load, capacity, and availability of quorum systems
- Naor, Wool
- 1998
(Show Context)
Citation Context ...and Chapter 8 of Grötschel et al., 1988), committees (Khachai et al., 2002), and quorum systems (intersecting set systems, intersecting hypergraphs) (Colbourn et al., 2001, Loeb and Conway, 2000, and =-=Naor and Wool, 1998-=-); see also Crama and Hammer (in preparation). The present paper is devoted to discussing questions concerning mechanisms of blocking in finite posets that go back to set-theoretic committees. We refe... |

44 |
Geometric Algorithms and
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(Show Context)
Citation Context ...,Gm} of nonempty subsets of a finite set if it holds |H∩Gk| > 0, for each k ∈ {1, . . .,m}. The family of all inclusion-minimal blocking sets for G is called the blocker of G, see, e.g., Chapter 8 of =-=Grötschel et al., 1993-=-. Let r be a rational number such that 0 ≤ r < 1. A set H Key words and phrases. Antichain, blocker, blocker map, clutter, committee, Farey sequence, lattice, poset. 2000 Mathematics Subject Classific... |

42 | Applications of basic hypergeometric functions - Andrews - 1974 |

27 |
Arrangements of Hyperplanes, Grundlehren der
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Citation Context ...⊂ Rn are linearly independent; 〈pi, x〉 := ∑n j=1 pijxj. 1≤i≤m Hi of the The connected components of the complement Rn − ⋃ hyperplane arrangement H are called the regions (or chambers) of H, see e.g., =-=Orlik and Terao, 1992-=-. We call the arrangement of oriented hyperplanes H (that is the set H for every hyperplane H of which “positive” and “negative sides” of H are distinguished) a training set, if a partition H = A ˙∪B ... |

26 | Algebraic Combinatorics on Convex Polytopes - Hibi - 1992 |

20 | Face numbers of polytopes and complexes
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- 1997
(Show Context)
Citation Context ...al antichain in B(n) then its order ideal I(A) is assigned the isomorphic face poset of the abstract simplicial complex whose facets are the sets from the family {I(a) ∩ B(n) (1) : a ∈ A}. See, e.g., =-=Billera and Björner, 1997-=-, Björner, 1995, Bruns and Herzog,8 ANDREY O. MATVEEV 1998, Buchstaber and Panov, 2004, Hibi, 1992, Miller and Sturmfels, 2004, Stanley, 1996, and Ziegler, 1998, on simplicial complexes. The followin... |

17 |
Combinatorial theory, Grundlehren der Mathematischen Wissenschaften
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- 1979
(Show Context)
Citation Context ...lled convex if the implication x, z ∈ C, y ∈ P, x ≤ y ≤ z in P =⇒ y ∈ C holds for all elements x, y, z ∈ P. We regard the empty subposet as a convex one. The Möbius function (see, e.g., Chapter IV of =-=Aigner, 1979-=-, Björner et al., 1997, Greene, 1982, and Chapter 3 of Stanley, 1997) µP : P ×P → Z is defined in the following way: µP(x, x) := 1, for any x ∈ P; further, if z ∈ P and x < z in P, then µP(x, z) := − ... |

13 |
Matchings and covers in hypergraphs
- Furedi
- 1988
(Show Context)
Citation Context ... cardinalities of finite sets and of their mutual intersections. Among mathematical constructions which underlie those procedures are blocking sets (covers, systems of representatives, transversals) (=-=Füredi, 1988-=-, and Chapter 8 of Grötschel et al., 1988), committees (Khachai et al., 2002), and quorum systems (intersecting set systems, intersecting hypergraphs) (Colbourn et al., 2001, Loeb and Conway, 2000, an... |

12 |
Tresser, A walk along the branches of the extended Farey tree
- Lagarias, P
- 1995
(Show Context)
Citation Context ... of all irreducible fractions between 0 and 1 whose denominators do not exceed n, see, e.g., Chapter 27 of Buchstab, 1967, Chapter 4 of Graham et al., 1994, Chapter III of Hardy and Wright, 1979, and =-=Lagarias and Tresser, 1995-=-. Thus, F(P, a; ω) is a subsequence of the Farey sequence of order ω(P). We always index the fractions from F(P, a; ω) starting with zero: F(P, a; ω) = ( f0 := 0 1 < f1 < f2 · · · < f|F(P,a;ω)|−1 := 1... |

10 | Subspace arrangements defined by products of linear forms, preprint, 2003, available at http://arXiv.org/abs/math/0401373
- Björner, Peeva, et al.
(Show Context)
Citation Context ...set, the set-theoretic concepts of blocking can be assigned order-theoretic counterparts. The next natural step consists in a passage from the Boolean lattices to arbitrary finite bounded posets, see =-=[7, 8, 25, 26, 27]-=-; a poset is called bounded if it has a least and greatest elements. Throughout the paper, P stands for a finite bounded poset of cardinality greater than one whose least and greatest elements are den... |

9 |
Note on a combinatorial application of Alexander duality
- Björner, Butler, et al.
- 1997
(Show Context)
Citation Context ... the implication x, z ∈ C, y ∈ P, x ≤ y ≤ z in P =⇒ y ∈ C holds for all elements x, y, z ∈ P. We regard the empty subposet as a convex one. The Möbius function (see, e.g., Chapter IV of Aigner, 1979, =-=Björner et al., 1997-=-, Greene, 1982, and Chapter 3 of Stanley, 1997) µP : P ×P → Z is defined in the following way: µP(x, x) := 1, for any x ∈ P; further, if z ∈ P and x < z in P, then µP(x, z) := − ∑ y∈P: x≤y<z µP(x, y);... |

9 |
Combinatorial Optimization: Packing and
- Cornuéjols
- 2001
(Show Context)
Citation Context ...nd this map assigns to a trivial clutter the other trivial clutter, see, e.g., Cordovil et al., 1991. The set-theoretic blocker constructions are at the foundation of discrete mathematics, see, e.g., =-=Cornuéjols, 2001-=-, and Crama and Hammer (in preparation). Since the clutters on a ground set are in one-to-one correspondence with the antichains in the Boolean lattice of all subsets of the ground set, the set-theore... |

9 |
Concrete Mathematics: A Foundation for
- Graham, Knuth, et al.
- 2002
(Show Context)
Citation Context ...quence Fn of order n ∈ P is defined to be the ascending sequence of all irreducible fractions between 0 and 1 whose denominators do not exceed n, see, e.g., Chapter 27 of Buchstab, 1967, Chapter 4 of =-=Graham et al., 1994-=-, Chapter III of Hardy and Wright, 1979, and Lagarias and Tresser, 1995. Thus, F(P, a; ω) is a subsequence of the Farey sequence of order ω(P). We always index the fractions from F(P, a; ω) starting w... |

9 |
Oriented matroids, Encyclopedia of Mathematics and its
- Björner, Vergnas, et al.
- 1993
(Show Context)
Citation Context ...the set of regions of H. The language of the theory of oriented matroids (which, for example, translates the regions of H to the maximal covectors of a realizable oriented matroid) may be of use; see =-=Björner et al., 1993-=-, on oriented matroids. Recall that the means of computing the rank of P, that is the number of regions of H, are well-known (Zaslavsky, 1975). Nonempty subsets of regions, regardedRELATIVE BLOCKING ... |

6 | Quorum systems constructed from combinatorial designs
- Colbourn, Dinitz, et al.
(Show Context)
Citation Context ... of representatives, transversals) (Füredi, 1988, and Chapter 8 of Grötschel et al., 1988), committees (Khachai et al., 2002), and quorum systems (intersecting set systems, intersecting hypergraphs) (=-=Colbourn et al., 2001-=-, Loeb and Conway, 2000, and Naor and Wool, 1998); see also Crama and Hammer (in preparation). The present paper is devoted to discussing questions concerning mechanisms of blocking in finite posets t... |

6 |
On the number of linear partitions of the (m
- Acketa, J
- 1991
(Show Context)
Citation Context ...F(B(n),m;ρ), for all m, 0 ≤ m ≤ n. See also Remark 3.3. Remark 7.10. In the present paper, we do not deal with the ascending Farey subsequences of the form ( h k ∈ Fn : h ≤ m ) , where 0 < m ≤ n (see =-=[1]-=- and Example 7.1,) including the classical Farey sequences Fn; see Section 6 for some references on Fn. Nevertheless, such Farey subsequences can be of use for the reader, and we list their basic prop... |

6 |
Matchings and covers in hypergraphs. Graphs and Combinatorics 4
- Füredi
- 1988
(Show Context)
Citation Context ... the cardinalities of finite sets and their mutual intersections. Among mathematical constructions which underlie those procedures are blocking sets (covers, systems of representatives, transversals) =-=[16]-=-, [19, Chapter 8], committees [22], and quorum systems (intersecting set systems, intersecting hypergraphs) [12, 24, 29]; see also [15]. The present paper is devoted to discussing questions concerning... |

5 |
Inconsistent homogeneous linear inequalities
- Ablow, Kaylor
- 1965
(Show Context)
Citation Context ...k ∈ Rk : 1 ≤ k ≤ t} is called a committee for the homogeneous system of strict linear inequalities {〈ei,x〉 > 0 : 1 ≤ i ≤ m}. Committees for such inequality systems were apparently first introduced in =-=Ablow and Kaylor, 1965-=-, where it was proved that such very useful collective generalizations of the notion of solution do exist. Those notes laid the foundation of a branch of the theory of pattern recognition; some of the... |

5 |
Simplicial matroids
- Cordovil, Lindström
- 1987
(Show Context)
Citation Context ...subset of the ground set, are called the trivial clutters. The blocker map assigns to a nontrivial clutter its blocker, and this map assigns to a trivial clutter the other trivial clutter, see, e.g., =-=Cordovil et al., 1991-=-. The set-theoretic blocker constructions are at the foundation of discrete mathematics, see, e.g., Cornuéjols, 2001, and Crama and Hammer (in preparation). Since the clutters on a ground set are in o... |

5 |
Voting fairly: transitive maximal intersecting families of sets, In memory of Gian-Carlo Rota
- Loeb, Conway
- 2000
(Show Context)
Citation Context ...ansversals) (Füredi, 1988, and Chapter 8 of Grötschel et al., 1988), committees (Khachai et al., 2002), and quorum systems (intersecting set systems, intersecting hypergraphs) (Colbourn et al., 2001, =-=Loeb and Conway, 2000-=-, and Naor and Wool, 1998); see also Crama and Hammer (in preparation). The present paper is devoted to discussing questions concerning mechanisms of blocking in finite posets that go back to set-theo... |

5 |
Metod Komitetov v Zadachakh Optimizatsii i Klassifikatsii
- Mazurov
(Show Context)
Citation Context ...e notes laid the foundation of a branch of the theory of pattern recognition; some of the surveys in the committee mathematical methods and their applications are Khachai, 2004, Khachai et al., 2002, =-=Mazurov, 1990-=-, Mazurov et al., 1989, and Mazurov and Khachai, 1999, 2004. The decision rule r is the mapping H → {−, +} under which r : H ↦→ λ(H); in other words, such a rule must correctly recognize the patterns ... |

4 |
A committee solution of the pattern recognition problem
- Ablow, Kaylor
- 1965
(Show Context)
Citation Context ...k ∈ Rk : 1 ≤ k ≤ t} is called a committee for the homogeneous system of strict linear inequalities {〈ei,x〉 > 0 : 1 ≤ i ≤ m}. Committees for such inequality systems were apparently first introduced in =-=Ablow and Kaylor, 1965-=-, where it was proved that such very useful collective generalizations of the notion of solution do exist. Those notes laid the foundation of a branch of the theory of pattern recognition; some of the... |

4 | On blockers in bounded posets
- Matveev
- 2001
(Show Context)
Citation Context ...of blocking can be assigned poset-theoretic counterparts. The next natural step consists in a passage from the Boolean lattices to arbitrary finite bounded posets, see Björner et al., 2004, 2005, and =-=Matveev, 2001-=-, 2002, 2003; a poset is called bounded if it has a least and greatest elements. Throughout the paper, P stands for a finite bounded poset of cardinality greater than one whose least and greatest elem... |

4 |
Committees of systems of linear inequalities, Automation and Remote Control
- Khachai
(Show Context)
Citation Context ...the notion of solution do exist. Those notes laid the foundation of a branch of the theory of pattern recognition; some of the surveys in the committee mathematical methods and their applications are =-=Khachai, 2004-=-, Khachai et al., 2002, Mazurov, 1990, Mazurov et al., 1989, and Mazurov and Khachai, 1999, 2004. The decision rule r is the mapping H → {−, +} under which r : H ↦→ λ(H); in other words, such a rule m... |

3 |
Toricheskie Deistviya v Topologii i Kombinatorike
- Buchstaber, Panov
(Show Context)
Citation Context ...f the abstract simplicial complex whose facets are the sets from the family {I(a) ∩ B(n) (1) : a ∈ A}. See, e.g., Billera and Björner, 1997, Björner, 1995, Bruns and Herzog,8 ANDREY O. MATVEEV 1998, =-=Buchstaber and Panov, 2004-=-, Hibi, 1992, Miller and Sturmfels, 2004, Stanley, 1996, and Ziegler, 1998, on simplicial complexes. The following proposition lists some observations. Proposition 2.3. holds (i) If A is a nontrivial ... |

3 |
Komitetnye Konstruktsii (in Russian) [Committee Constructions
- Khachai
- 1999
(Show Context)
Citation Context ... of pattern recognition; some of the surveys in the committee mathematical methods and their applications are Khachai, 2004, Khachai et al., 2002, Mazurov, 1990, Mazurov et al., 1989, and Mazurov and =-=Khachai, 1999-=-, 2004. The decision rule r is the mapping H → {−, +} under which r : H ↦→ λ(H); in other words, such a rule must correctly recognize the patterns from the training set. Given a committee {wk : 1 ≤ k ... |

3 |
Incidence Functions as Generalized Arithmetic Functions, I
- Smith
- 1967
(Show Context)
Citation Context ...iated to nonempty antichains A in P; such sequences can also be of interest. Order-preserving maps P → P and P → P ∗ , where P ∗ are positive integers ordered by divisibility, are discussed, e.g., in =-=Smith, 1967-=-, 1969, 1970/1971. See Pǎtra¸scu and Pǎtra¸scu, 2004, on algorithmic aspects of the Farey sequences. 7. Farey subsequences in Boolean context In this section we deal almost exclusively with the Boolea... |

3 | Komitetnye Resheniya Nesovmestnykh Sistem Ogranichenii i Metody Obucheniya Raspoznavaniyu (in Russian) [Committee Solutions of Infeasible - Khachai - 2004 |

2 |
Cohen-Macaulay Rings, Second edition, Cambridge
- Bruns, Herzog
- 1998
(Show Context)
Citation Context ...al antichain in B(n) then its order ideal I(A) is assigned the isomorphic face poset of the abstract simplicial complex whose facets are the sets from the family {I(a) ∩ B(n) (1) : a ∈ A}. See, e.g., =-=[4, 5, 9, 11, 21, 28, 32, 34]-=- on simplicial complexes. The following proposition lists some observations. Proposition 2.3. holds (i) If A is a nontrivial antichain in P, then it Ir(P, A; ω) = ⋂ Ir(P, a; ω) , a∈A for any map ω. (i... |

2 |
The Möbius function of a partially ordered set, in: Ordered Sets
- Greene
- 1981
(Show Context)
Citation Context ...d convex if the implication x, z ∈ C, y ∈ P, x ≤ y ≤ z in P =⇒ y ∈ C holds for all elements x, y, z ∈ P. We regard the empty subposet as a convex one. The Möbius function (see, e.g., [2, Chapter IV], =-=[6, 18]-=-, [33, Chapter 3]) µP : P × P → Z is defined in the following way: µP(x, x) := 1, for any x ∈ P; further, if z ∈ P and x < z in P, then µP(x, z) := − ∑ y∈P: x≤y<z µP(x, y); finally, if x ≰ z in P, the... |

2 |
A note on operators of deletion and contraction for antichains
- Matveev
(Show Context)
Citation Context ...set, the set-theoretic concepts of blocking can be assigned order-theoretic counterparts. The next natural step consists in a passage from the Boolean lattices to arbitrary finite bounded posets, see =-=[7, 8, 25, 26, 27]-=-; a poset is called bounded if it has a least and greatest elements. Throughout the paper, P stands for a finite bounded poset of cardinality greater than one whose least and greatest elements are den... |

2 |
Extended blocker, deletion, and contraction maps on antichains
- Matveev
(Show Context)
Citation Context ...(A) by definition is the other trivial antichain. The antichains bj(A), defined by (1.1), serve as a poset-theoretic generalization of the notion of set-theoretic blocker of a nontrivial clutter, see =-=Matveev, 2003-=-. From this point of view, the antichain b(A) := b0(A) (1.2)RELATIVE BLOCKING IN POSETS 3 bears a strong resemblance to its set-theoretic predecessor, see Björner and Hultman, 2004, and Matveev, 2001... |

2 | A note on blockers in posets
- Björner, Hultman
(Show Context)
Citation Context ...notion of set-theoretic blocker of a nontrivial clutter, see Matveev, 2003. From this point of view, the antichain b(A) := b0(A) (1.2) bears a strong resemblance to its set-theoretic predecessor, see =-=Björner and Hultman, 2004-=-, and Matveev, 2001. Antichains (1.1) admit a niceRELATIVE BLOCKING IN POSETS 3 ordering, and some of the structural and combinatorial properties of blockers (1.2) in the Boolean lattices are clarifi... |

2 |
with contributions by
- Crama, Hammer
(Show Context)
Citation Context ...blocking sets (covers, systems of representatives, transversals) [16], [19, Chapter 8], committees [22], and quorum systems (intersecting set systems, intersecting hypergraphs) [12, 24, 29]; see also =-=[15]-=-. The present paper is devoted to discussing questions concerning mechanisms of blocking in finite posets that go back to set-theoretic committees. We refer the reader to [33, Chapter 3] for informati... |

2 |
Lectures on Polytopes, Second edition, Graduate Texts in Mathematics
- Ziegler
- 1998
(Show Context)
Citation Context ...al antichain in B(n) then its order ideal I(A) is assigned the isomorphic face poset of the abstract simplicial complex whose facets are the sets from the family {I(a) ∩ B(n) (1) : a ∈ A}. See, e.g., =-=[4, 5, 9, 11, 21, 28, 32, 34]-=- on simplicial complexes. The following proposition lists some observations. Proposition 2.3. holds (i) If A is a nontrivial antichain in P, then it Ir(P, A; ω) = ⋂ Ir(P, a; ω) , a∈A for any map ω. (i... |

1 | On the number of linear partitions of the (m,n)grid - Acketa, J - 1991 |

1 | Teoria Chisel (in Russian) [Number Theory] Uchpedgiz - Buchstab - 1960 |

1 | Prognoz [Pattern Recognition. Classification - Kassifikatsiya - 1989 |

1 | Computing order statistics in the Farey sequence, in
- Pǎtra¸scu, Pǎtra¸scu
(Show Context)
Citation Context ...pty antichains A in P; such sequences can also be of interest. Order-preserving maps P → P and P → P ∗ , where P ∗ are positive integers ordered by divisibility, are discussed, e.g., in [32, 33]. See =-=[31]-=- on algorithmic aspects of Farey sequences. 7. Farey subsequences in Boolean context In this section we deal almost exclusively with the Boolean lattice B(n). Let a be an arbitrary element of B(n), of... |

1 |
Multiplication operators on incidence algebras
- Smith
(Show Context)
Citation Context ... with nonempty antichains A in P; such sequences can also be of interest. Order-preserving maps P → P and P → P ∗ , where P ∗ are nonnegative integers ordered by divisibility, are discussed, e.g., in =-=[30, 31]-=-. 7. Farey subsequences in Boolean context In this section we deal almost exclusively with the Boolean lattice B(n). Let a be an arbitrary element of B(n), of rank m := ρ(a). Consider the Farey subseq... |

1 | Teoria Chisel - Buchstab - 1960 |

1 |
Committee constructions for solving problems of selection, diagnostics, and prediction
- Khachai, Rybin
(Show Context)
Citation Context ...and their mutual intersections. Among mathematical constructions which underlie those procedures are blocking sets (covers, systems of representatives, transversals) [16], [19, Chapter 8], committees =-=[22]-=-, and quorum systems (intersecting set systems, intersecting hypergraphs) [12, 24, 29]; see also [15]. The present paper is devoted to discussing questions concerning mechanisms of blocking in finite ... |

1 |
Maps on posets, and blockers, preprint arXiv:math.CO/0411025
- Matveev
(Show Context)
Citation Context ...set, the set-theoretic concepts of blocking can be assigned poset-theoretic counterparts. The next natural step consists in a passage from the Boolean lattices to arbitrary finite bounded posets, see =-=[7, 8, 25, 26, 27, 28]-=-; a poset is called bounded if it has a least and greatest elements. Throughout the paper, P stands for a finite bounded poset of cardinality greater than one whose least and greatest elements are den... |