ON THE COMPLEXITY OF COMPUTING VALUES OF RESTRICTED GAMES

2002 ◽  
Vol 13 (05) ◽  
pp. 633-651 ◽  
Author(s):  
J. M. BILBAO ◽  
J. R. FERNÁNDEZ ◽  
J. J. LÓPEZ
Keyword(s):  
Cooperative Games ◽  
Partially Ordered Set ◽  
Polynomial Algorithm ◽  
Ordered Set ◽  
Combinatorial Structure ◽  
Distributive Lattices ◽  
Communication Graph ◽  
Partially Ordered ◽  
Connected Subgraphs ◽  
Restricted Games

The aim of this paper is to compute Shapley's and Banzhaf's values of cooperative games restricted by a combinatorial structure. There have been previous models developed to study the problem of games with partial cooperation. Games restricted by a communication graph were introduced by Myerson and Owen. Another type of combinatorial structure introduced by Gilles, Owen and van den Brink is equivalent to a subclass of antimatroids. Cooperative games in which the set of players is a partially ordered set, that is, games on distributive lattices was investigated by Faigle and Kern. We introduce a new combinatorial structure called augmenting system which is a generaligation of the antimatroid structure and the system of connected subgraphs of graph. We present new algorithmic procedures for computing values of games under augmenting systems restrictions and we show that there exist problems with polynomial algorithm complexity.

On the Partially Ordered Set of Prime Ideals of a Distributive Lattice

1971 ◽  
Vol 23 (5) ◽  
pp. 866-874 ◽  
Author(s):  
Raymond Balbes
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Complete Characterization ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Partially Ordered ◽  
Irreducible Elements

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.

scholarly journals Finitely generated pseudocomplemented distributive lattices

1975 ◽  
Vol 19 (2) ◽  
pp. 238-246 ◽  
Author(s):  
J. Berman ◽  
ph. Dwinger
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Distributive Lattices ◽  
Basic Fact ◽  
Finitely Generated ◽  
Partially Ordered ◽  
Finite Set ◽  
Natural Way

If L is a pseudocomplemented distributive lattice which is generated by a finite set X, then we will show that there exists a subset G of L which is associated with X in a natural way that ¦G¦ ≦ ¦X¦ + 2¦x¦ and whose structure as a partially ordered set characterizes the structure of L to a great extent. We first prove in Section 2 as a basic fact that each element of L can be obtained by forming sums (joins) and products (meets) of elements of G only. Thus, L considered as a distributive lattice with 0,1 (the operation of pseudocomplementation deleted), is generated by G. We apply this to characterize for example, the maximal homomorphic images of L in each of the equational subclasses of the class Bω of pseudocomplemented distributive lattices, and also to find the conditions which have to be satisfied by G in order that X freely generates L.

Information Retrieval Systems, an Algebraic Approach I1

1981 ◽  
Vol 4 (3) ◽  
pp. 551-603
Author(s):  
Zbigniew Raś
Keyword(s):  
Information Retrieval ◽  
Boolean Algebra ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Algebraic Approach ◽  
Partially Ordered ◽  
Retrieval Systems ◽  
Information Retrieval Systems ◽  
Present Part ◽  
L Systems

This paper is the first of the three parts of work on the information retrieval systems proposed by Salton (see [24]). The system is defined by the notions of a partially ordered set of requests (A, ⩽), the set of objects X and a monotonic retrieval function U : A → 2X. Different conditions imposed on the set A and a function U make it possible to obtain various classes of information retrieval systems. We will investigate systems in which (A, ⩽) is a partially ordered set, a lattice, a pseudo-Boolean algebra and Boolean algebra. In my paper these systems are called partially ordered information retrieval systems (po-systems) lattice information retrieval systems (l-systems); pseudo-Boolean information retrieval systems (pB-systems) and Boolean information retrieval systems (B-systems). The first part concerns po-systems and 1-systems. The second part deals with pB-systems and B-systems. In the third part, systems with a partial access are investigated. The present part discusses the method for construction of a set of attributes. Problems connected with the selectivity and minimalization of a set of attributes are investigated. The characterization and the properties of a set of attributes are given.

scholarly journals Each join-completion of a partially ordered set in the solution of a universal problem

1974 ◽  
Vol 17 (4) ◽  
pp. 406-413 ◽  
Author(s):  
Jürgen Schmidt
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Unique Extension ◽  
Partially Ordered ◽  
Special Cases

The main result of this paper is the theorem in the title. Only special cases of it seem to be known so far. As an application, we obtain a result on the unique extension of Galois connexions. As a matter of fact, it is only by the use of Galois connexions that we obtain the main result, in its present generality.

scholarly journals Primitive idempotent measures on compact semitopological semigroups

1972 ◽  
Vol 13 (4) ◽  
pp. 451-455 ◽  
Author(s):  
Stephen T. L. Choy
Keyword(s):  
Partial Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Zero Element ◽  
Primitive Idempotent ◽  
Natural Partial Order ◽  
Partially Ordered

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

ON RIBBON PRESENTATIONS OF RIBBON KNOTS

1994 ◽  
Vol 03 (02) ◽  
pp. 223-231
Author(s):  
TOMOYUKI YASUDA
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Totally Ordered Set

A ribbon n-knot Kn is constructed by attaching m bands to m + 1n-spheres in the Euclidean (n + 2)-space. There are many way of attaching them; as a result, Kn has many presentations which are called ribbon presentations. In this note, we will induce a notion to classify ribbon presentations for ribbon n-knots of m-fusions (m ≥ 1, n ≥ 2), and show that such classes form a totally ordered set in the case of m = 2 and a partially ordered set in the case of m ≥ 1.

scholarly journals Spherical posets from commuting elements

2018 ◽  
Vol 21 (4) ◽  
pp. 593-628 ◽  
Author(s):  
Cihan Okay
Keyword(s):  
Homotopy Type ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Universal Cover ◽  
Homotopy Equivalent ◽  
Classifying Space ◽  
Partially Ordered ◽  
Abelian Subgroups

AbstractIn this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where {2r\geq 4} is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

EMBEDDINGS OF P(ω)/Fin INTO BOREL EQUIVALENCE RELATIONS BETWEEN ℓp AND ℓq

2015 ◽  
Vol 80 (3) ◽  
pp. 917-939 ◽  
Author(s):  
ZHI YIN
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Partially Ordered

AbstractWe prove that, for 1 ≤ p < q < ∞, the partially ordered set P(ω)/Fin can be embedded into Borel equivalence relations between ℝω/ℓp and ℝω/ℓq. Since there is an antichain of size continuum in P(ω)/Fin, there are continuum many pairwise incomparable Borel equivalence relations between ℝω/ℓp and ℝω/ℓq.

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