Showing papers in "Order in 2005"
TL;DR: It is proved the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution.
Abstract: We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets.
1,159 citations
TL;DR: The variety of semirings generated by all ordered bands is found and part of the lattice of its subvarieties is determined.
Abstract: The lattice of all subvarieties of the variety generated by all ordered bands is obtained. This lattice is distributive and contains 78 varieties precisely. Each of these is finitely based and generated by a finite number of finite ordered bands.
49 citations
TL;DR: If G has an orientation with maximum outdegree k then the (k, 1)-game coloring number of G is at most 2k + 2.5, which extends a fundamental result on the (1, 1), which is tight for many classes of graphs.
Abstract: We investigate a competitive version of the coloring number of a graph G = (V, E) For a fixed linear ordering L of V let s (L) be one more than the maximum outdegree of G when G is oriented so that x ← y if x
38 citations
TL;DR: It will be proved that a continuous analogue of the Szpilrajn theorem does not hold in general and necessary and in some cases necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue for the Dushnik-Miller theorem will be presented.
Abstract: The Szpilrajn theorem and its strengthening by Dushnik and Miller belong to the most quoted theorems in many fields of pure and applied mathematics as, for instance, order theory, mathematical logic, computer sciences, mathematical social sciences, mathematical economics, computability theory and fuzzy mathematics. The Szpilrajn theorem states that every partial order can be refined or extended to a total (linear) order. The theorem by Dushnik and Miller states, moreover, that every partial order is the intersection of its total (linear) refinements or extensions. Since in mathematical social sciences or, more general, in any theory that combines the concepts of topology and order one is mainly interested in continuous total orders or preorders in this paper some aspects of a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller will be discussed. In particular, necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Dushnik-Miller theorem will be presented. In addition, it will be proved that a continuous analogue of the Szpilrajn theorem does not hold in general. Further, necessary and in some cases necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Szpilrajn theorem will be presented.
37 citations
TL;DR: It is proved that the prism over Bk is hamiltonian and it is shown that Bk has a closed spanning 2-trail.
Abstract: Let Bk be the bipartite graph defined by the subsets of {1,…,2k + 1} of size k and k + 1. We prove that the prism over Bk is hamiltonian. We also show that Bk has a closed spanning 2-trail.
34 citations
TL;DR: A lower bound of the linear discrepancy is given and then injective isotones on the product of two chains are constructed, which show that the obtained lower bound is tight.
Abstract: The linear discrepancy of a partially ordered set P = (X, ≺) is the minimum integer l such that ∣f(a) − f(b)∣ ≤ l for any injective isotone $$
f:P \to \mathbb{Z}
$$
and any pair of incomparable elements a, b in X. It measures the degree of difference of P from a chain. Despite of increasing demands to the applications, the discrepancies of just few simple partially ordered sets are known. In this paper, we obtain the linear discrepancy of the product of two chains. For this, we firstly give a lower bound of the linear discrepancy and then we construct injective isotones on the product of two chains, which show that the obtained lower bound is tight.
33 citations
TL;DR: It is proved that the lattices of lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.
Abstract: In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference
135 (2005), 77–92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math.
217 (2000), 367–409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.
22 citations
TL;DR: This work focuses on the generating cone of supermodular functions because the extreme rays of that cone can be used as test functions to determine whether two random variables are ordered under thesupermodular order.
Abstract: The supermodular order on multivariate distributions has many applications in financial and actuarial mathematics. In the particular case of finite, discrete distributions, we generalize the order to distributions on finite lattices. In this setting, we focus on the generating cone of supermodular functions because the extreme rays of that cone (modulo the modular functions) can be used as test functions to determine whether two random variables are ordered under the supermodular order. We completely determine the extreme supermodular functions in some special cases.
15 citations
TL;DR: It is shown that a section semi-complemented pseudocomplemented poset is a Boolean poset and Peirce's Theorem is extended to uniquely complemented posets (with 0 and 1) like Peirces' Theorem and the Birkhoff–von Neumann Theorem.
Abstract: In this paper, some classical results of uniquely complemented lattices are extended to uniquely complemented posets (with 0 and 1) like Peirce's Theorem, the Birkhoff–von Neumann Theorem, the Birkhoff–Ward Theorem. Further, it is shown that a section semi-complemented pseudocomplemented poset is a Boolean poset.
13 citations
TL;DR: It is shown that pseudotrees are precisely those posets for which consistent sets, directed sets, and nonempty chains coincide, and it is proved that Pseudo-Chained Posets are preciselyThose posets that admit a set representation by sets of appropriate chains.
Abstract: This paper provides new results on pseudotrees. First, it is shown that pseudotrees are precisely those posets for which consistent sets, directed sets, and nonempty chains coincide. Second, we show that chain-complete pseudotrees yield complete meet-semilattices. Third, we prove that pseudotrees are precisely those posets that admit a set representation by sets of appropriate chains. This latter result generalizes results needed for applications in game theory and economics.
9 citations
TL;DR: Some examples of Σ11-universal preorders are presented, in the form of various relations of embeddability between countable coloured total orders, to strengthen a theorem of 11-universality of continuous embeddable for dendrites whose branch points have order 3.
Abstract: Some examples of Σ11-universal preorders are presented, in the form of various relations of embeddability between countable coloured total orders. As an application, strengthening a theorem of (Marcone, A. and Rosendal, C.: The Complexity of Continuous Embeddability between Dendrites, J. Symb. Log.69 (2004), 663–673), the Σ11-universality of continuous embeddability for dendrites whose branch points have order 3 is obtained.
TL;DR: It is proved that a partial order can be linearly extended if and only if it can beLinearly extended on every finitely generated subalgebra.
Abstract: We answer the question, when a partial order in a partially ordered algebraic structure has a compatible linear extension. The finite extension property enables us to show, that if there is no such extension, then it is caused by a certain finite subset in the direct square of the base set. As a consequence, we prove that a partial order can be linearly extended if and only if it can be linearly extended on every finitely generated subalgebra. Using a special equivalence relation on the above direct square, we obtain a further property of linearly extendible partial orders. Imposing conditions on the lattice of compatible quasi orders, the number of linear orders can be determined. Our general approach yields new results even in the case of semi-groups and groups.
TL;DR: This work derives exact recursive formulas for the number of homomorphisms between two related types of weakly ordered sets and proves a strong automorphism conjecture for series-parallel ordered sets.
Abstract: The automorphism conjecture for ordered sets states that the automorphism to endomorphism ratio will tend to zero as the size of the ordered set goes to infinity. We show by computer enumeration that up to size 11 the ratio is largest for weakly ordered sets. Subsequently, we derive exact recursive formulas for the number of homomorphisms between two related types of weakly ordered sets and we prove a strong automorphism conjecture for series-parallel ordered sets. We conclude with an example that shows that the automorphism to endomorphism ratio can exceed \(\displaystyle{ \left( .5244 \right) ^{|P|}} \) for arbitrarily large \(|P|\).
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TL;DR: The notion of a homomorphism, of a congruence relation, of an substructure of a poset and consequently the notion ofA variety of posets is introduced and investigated.
Abstract: We introduce and investigate the notion of a homomorphism, of a congruence relation, of a substructure of a poset and consequently the notion of a variety of posets. These notions are consistent with those used in lattice theory and multilattice theory. There are given some properties of the lattice of all varieties of posets.
TL;DR: This work investigates what type of number-valued function is induces a ∧-subsemilattice of the type of $$\mathcal{L}$$ (or cl) and if so, what kind of
Abstract: Let $$\mathcal{D}$$
be a lattice of finite height. The correspondence between closure operators $$cl:\mathcal{D} \to \mathcal{D}$$
and ∧-subsemilattices $$\mathcal{L} \subseteq \mathcal{D}$$
is well known. Here we investigate what type of number-valued function $$\mathcal{D} \to \mathbb{N}$$
is induces a ∧-subsemilattice $$\mathcal{L}$$
; and if so, what kind of $$\mathcal{L}$$
. Conversely, what type of function $$\mathcal{D} \to \mathbb{N}$$
is induced by what type of $$\mathcal{L}$$
(or cl). Several results known for matroids, greedoids, or semimodular lattices are generalized.
TL;DR: Any topology on X is generated by a partial pseudometric generated by any preorder on a finite set X if and only if p(a,b) \leq p (a,a)$$.
Abstract: Given any preorder \(\preceq\) on a finite set \(X\), we present an algorithm to construct a partial pseudometric \(p\) on \(X\) which generates \(\preceq\) in the sense that \(a \preceq b\) if and only if \(p(a,b) \leq p(a,a)\). The specialization topology generated by \(\preceq\) agrees with the topology generated by the partial pseudometric \(p\)-balls, and consequently any topology on \(X\) is generated by a partial pseudometric.
TL;DR: Minimal representations for several important functions of bounded variation on several of the posets mentioned above are determined in this paper.
Abstract: We study the so-called Skorokhod reflection problem (SRP) posed for real-valued functions defined on a partially ordered set (poset), when there are two boundaries, considered also to be functions of the poset. The problem is to constrain the function between the boundaries by adding and subtracting nonnegative nondecreasing (NN) functions in the most efficient way. We show existence and uniqueness of its solution by using only order theoretic arguments. The solution is also shown to obey a fixed point equation. When the underlying poset is a σ-algebra of subsets of a set, our results yield a generalization of the classical Jordan–Hahn decomposition of a signed measure. We also study the problem on a poset that has the structure of a tree, where we identify additional structural properties of the solution, and on discrete posets, where we show that the fixed point equation uniquely characterizes the solution. Further interesting posets we consider are the poset of real n-vectors ordered by majorization, and the poset of n × n positive semidefinite real matrices ordered by pointwise ordering of the associated quadratic forms. We say a function on a poset is of bounded variation if it can be written as the difference of two NN functions. The solution to the SRP when the upper and lower boundaries are the identically zero function corresponds to the most efficient or minimal such representation of a function of bounded variation. Minimal representations for several important functions of bounded variation on several of the posets mentioned above are determined in this paper.
TL;DR: Kubicki, Lehel and Morayne have proved the conjecture that when T1, T2 are restricted to being binary trees, the corresponding correlation inequalities for these maps also generalise to arbitrary trees.
Abstract: Let Tn be the complete binary tree of height n, with root 1n as the maximum element. For T a tree, define \(A(n;T) = \vert{ \{S \subseteq T^{n} : 1_{n} \in S, S \cong T\} \vert}\) and \(C(n;T) = \vert{ \{S \subseteq T^{n} : S \cong T\} \vert}\). We disprove a conjecture of Kubicki, Lehel and Morayne, which claims that \(\frac{A(n;T_1)}{C(n;T_1)} \leq \frac{A(n;T_2)}{C(n;T_2)}\) for any fixed n and arbitrary rooted trees T1T2. We show that A(n; T) is of the form \(\sum_{j=0}^lq_j(n) 2^{jn}\) where l is the number of leaves of T, and each qj is a polynomial. We provide an algorithm for calculating the two leading terms of ql for any tree T. We investigate the asymptotic behaviour of the ratio A(n; T)/C(n; T) and give examples of classes of pairs of trees T1, T2 where it is possible to compare A(n; T1)/C(n; T1) and A(n; T2)/C(n; T2). By calculating these ratios for a particular class of pairs of trees, we show that the conjecture fails for these trees, for all sufficiently large n. Kubicki, Lehel and Morayne have proved the conjecture when T1, T2 are restricted to being binary trees. We also look at embeddings into other complete trees, and we show how the result can be viewed as one of many possible correlation inequalities for embeddings of binary trees. We also show that if we consider strict order-preserving maps of T1, T2 into Tn (rather than embeddings) then the corresponding correlation inequalities for these maps also generalise to arbitrary trees.
[...]
TL;DR: Given a poset, the notion of Q-upper algebras is introduced and the (positive) implicativity, commutativity and quasi-commutativity in Q- Upper algebraes are studied.
Abstract: Given a poset we introduce the notion of Q-upper algebras and study the (positive) implicativity, commutativity and quasi-commutativity in Q-upper algebras.
TL;DR: A global order of norms using strongly optimal strategies in Blackwell games is defined and it is proved that it is a prewellordering under the assumption of the Axiom of Blackwell determinacy.
Abstract: We define a global order of norms using strongly optimal strategies in Blackwell games and prove that it is a prewellordering under the assumption of the Axiom of Blackwell determinacy.
TL;DR: Algebraic properties of order-primal algebras for connected ordered sets (A; ≤) are studied and several properties of the varieties and the quasi-varieties generated by constantive and simple algebraes are applied.
Abstract: A finite algebra \(\underline{A} = (A; F^{\underline{A}})\) is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order ≤ on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; ≤). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case \(F^{\underline A}\) cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras.
TL;DR: An isometric sublattice on a six-dimensional vector space lattice is constructed having a congruence relation such that the factor lattice does not admit any representation via contraction.
Abstract: An isometric sublattice on a six-dimensional vector space lattice is constructed having a congruence relation such that the factor lattice does not admit any representation via contraction.
TL;DR: A point-free version of the classical Kakutani duality is described and an adjunction is found between the category of compact completely regular frames with frame maps and the categories of Archimedean bounded Riesz spaces with continuous RiesZ maps.
Abstract: There are many results proved using the Axiom of Choice. Using point-free topology, we can prove some of these results without using this axiom. B. Banaschewski in [Pointfree Topology and the Spectra of f-rings, Ordered algebraic structures (Curacoa, 1995), Kluwer, Dordrecht, 123–148], studying the spectra of f-rings, describes the point-free version of the classical Gelfand duality without using the Axiom of Choice In this paper, referring to [Ebrahimi, M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct.12 (2004), 397–409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of l-Modules, To appear in J. Pure Appl. Algebra], we describe a point-free version of the classical Kakutani duality. For this, using one of the spectra given in [Ebrahimi, M. M., Karimi Feizabadi, A. and Mahmoudi, M.: Pointfree Spectra of Riesz Space, Appl. Categ. Struct.12 (2004), 397–409; Ebrahimi, M. M. and Karimi Feizabadi, A.: Pointfree Spectra of l-Modules, To appear in J. Pure Appl. Algebra], we find an adjunction between the category of compact completely regular frames with frame maps and the category of Archimedean bounded Riesz spaces with continuous Riesz maps.
TL;DR: In this paper, a class of ranked posets (Dhk, ≪) has been defined in order to analyse, from a combinatorial viewpoint, particular systems of real homogeneous inequalities between monomials.
Abstract: A class of ranked posets {(Dhk, ≪)} has been recently defined in order to analyse, from a combinatorial viewpoint, particular systems of real homogeneous inequalities between monomials. In the present paper we focus on the posets D2k, which are related to systems of the form {xaxb *abcdxcxd: 0 ≤ a, b, c, d ≤ k, *abcd ∈ { }, 0 < x0 < x1 < ...< xk}. As a consequence of the general theory, the logical dependency among inequalities is adequately captured by the so-defined posets \({\left( {{\user1{\mathcal{W}}}^{k}_{2} , < } \right)}\). These structures, whose elements are all the D2k's incomparable pairs, are thoroughly surveyed in the following pages. In particular, their order ideals – crucially significant in connection with logical consequence – are characterised in a rather simple way. In the second part of the paper, a class of antichains \({\left\{ {{\user1{{\wp }}}_{k} \subseteq {\user1{\mathcal{W}}}^{k}_{2} } \right\}}\) is shown to enjoy some arithmetical properties which make it an efficient tool for detecting incompatible systems, as well as for posing some compatibility questions in a purely combinatorial fashion.
[...]
TL;DR: It is shown that certain partially ordered rings, defined by some of the properties of the totally ordered ring of integers, are exactly the bounded Z-rings, that is, the commutative f-rings with strong singular unit.
Abstract: It is shown that certain partially ordered rings, defined by some of the properties of the totally ordered ring of integers, are exactly the bounded Z-rings, that is, the commutative f-rings with strong singular unit. The partially ordered rings in question amount to a discrete version of the rings introduced by M.H. Stone for his abstract characterization of the rings of real-valued continuous functions on compact Hausdorff spaces, and the function rings they correspond to are given by the integer-valued continuous functions on Boolean spaces.