Incomparable Elements Research Articles

Drawing Order Diagrams Through Two-Dimension Extension

Ordinal real-world data such as concept hierarchies, ontologies, genealogies, or task dependencies in scheduling often has the property to not only contain pairwise comparable, but also incomparable elements. Order diagrams provide an important tool for understanding and thus discovering knowledge in such data. Easily readable drawings of such order diagrams are hard to come by, even for small ordered sets. Many attempts were made to transfer classical graph drawing approaches to order diagrams. Although these methods produce satisfying results for some ordered sets, they unfortunately perform poorly in general. In this work, we present the novel algorithm DimDraw to decompose an ordered set (e.g., a concept hierarchy) in linear orders and to produce a corresponding order diagram. This algorithm is based on a relation between the dimension of an ordered set and the bipartiteness of its transitive incompatibility graph. To evaluate the quality of the algorithm, a user study was conducted where generated drawings were compared with ones from state-of-the-art drawing algorithms.

Shortened universal cycles for permutations

Kitaev et al. (2019) described how to shorten universal words for permutations, to length n ! + n − 1 − i ( n − 1 ) for any i ∈ [ ( n − 2 ) ! ] , by introducing incomparable elements. They conjectured that it is also possible to use incomparable elements to shorten universal cycles for permutations to length n ! − i ( n − 1 ) for any i ∈ [ ( n − 2 ) ! ] . In this note we prove their conjecture. The proof is constructive, and, on the way, we also show a new method for constructing universal cycles for permutations.

Construction of Uninorms on Bounded Lattices with Incomparable Elements that are Neither Conjunctive Nor Disjunctive

This paper investigates uninorms that are neither conjunctive nor disjunctive on bounded lattices. New methods are introduced for construction of such uninorms, where some restrictions on the identity and the annihilator are considered. In particular, new types of idempotent uninorms on bounded lattices are obtained. Furthermore, some specific examples are provided to illustrate that these constructions differ from the existing ones.

Estimating and testing relations and trees on the basis of pairwise comparisons

Abstract The paper presents an ample account on author’s results, concerning the methods of estimating and testing of relations (equivalence, tolerance, preference) and trees on the basis of multiple pairwise comparisons with random errors, which are presented in a new book, Klukowski (2021c). The book contains also main results presented earlier in Klukowski (2011). The new content pertains mainly to: the preference relation with incomparable elements, trees and tests for verification of estimates of the relations. The estimators and tests have good statistical properties; estimators - consistency with good convergence, tests - known distributions of test statistics (exact, limiting or approximate). Some properties of estimates (precision and convergence) have been determined using the simulation approach. The results here reported gain in significance in view of the renewed interest in practical application of pairwise comparisons in various application domains, which give rise to various concrete data analysis problems.

Lazy Queue Layouts of Posets

We investigate the queue number of posets in terms of their width, that is, the maximum number of pairwise incomparable elements. A long-standing conjecture of Heath and Pemmaraju asserts that every poset of width w has queue number at most w. The conjecture has been confirmed for posets of width w=2 via so-called lazy linear extension. We extend and thoroughly analyze lazy linear extensions for posets of width w > 2. Our analysis implies an upper bound of (w-1)^2 +1 on the queue number of width-w posets, which is tight for the strategy and yields an improvement over the previously best-known bound. Further, we provide an example of a poset that requires at least w+1 queues in every linear extension, thereby disproving the conjecture for posets of width w > 2.

A property of meets in slim semimodular lattices and its application to retracts

Slim semimodular lattices were introduced by G. Grätzer and E. Knapp in 2007, and they have intensively been studied since then. These lattices can be given by \(\mathcal{C}_1\)-diagrams, defined by the author in 2017. We prove that if x and y are incomparable elements in such a lattice L, then their meet has the property that the interval \([x \wedge y, x]\) is a chain, this chain is of a normal slope in every \(\mathcal{C}_1\)-diagram of L, and except possibly for x, the elements of this chain are meet-reducible.In the direct square K1 of the three-element chain, let X1 and A1 be the set of atoms and the sublattice generated by 0 and the coatoms, respectively. Denote by K2 the unique eight-element lattice embeddable in K1. Let A2 be the sublattice of K2 consisting of 0, 1, the meet-reducible atom, and the join-reducible coatom. Let X2 stand for the singleton consisting of the doubly reducible element of K2. For i = 1, 2, we apply the above-mentioned property of meets to prove that whenever Ki is a sublattice and Si is a retract of a slim semimodular lattice, then \(A_i \subseteq S_i\) implies that \(X_i \subseteq S_i\).

Induced betweenness in order-theoretic trees

The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x<y<z∨z<y<x, and the corresponding betweenness structure is (N,B). The class of betweenness structures of linear orders is first-order definable. That of partial orders is monadic second-order definable. An order-theoretic tree is a partial order such that the set of elements larger that any element is linearly ordered and any two elements have an upper-bound. Finite or infinite rooted trees ordered by the ancestor relation are order-theoretic trees. In an order-theoretic tree, B(x,y,z) means that x<y<z or z<y<x or x<y≤x⊔z or z<y≤x⊔z, where x⊔z is the least upper-bound of incomparable elements x and z. In a previous article, we established that the corresponding class of betweenness structures is monadic second-order definable.We prove here that the induced substructures of the betweenness structures of the countable order-theoretic trees form a monadic second-order definable class, denoted by IBO. The proof uses a variant of cographs, the partitioned probe cographs, and their known six finite minimal excluded induced subgraphs called the bounds of the class. This proof links two apparently unrelated topics: cographs and order-theoretic trees.However, the class IBO has finitely many bounds, i.e., minimal excluded finite induced substructures. Hence it is first-order definable. The proof of finiteness uses well-quasi-orders and does not provide the finite list of bounds. Hence, the associated first-order defining sentence is not known.

The symmetric group, ordered by refinement of cycles, is strongly Sperner

Given a poset $(P,\leq )$, an antichain is a subset of pairwise incomparable elements of $P$. Let $(P,w)$ be a graded, weighted poset. If the maximum weight of an antichain of $P$ is equal to the weight of the largest rank of $P$, then $P$ is said to be Sperner. In 1967, Rota conjectured that the poset of partitions, ordered by refinement of blocks, is Sperner; this conjecture was later disproved by Canfield. In this paper, we consider a generalization of Rota’s conjecture and show that $S_n$, partially ordered by refinement of cycles, is strongly Sperner.

Общая характеристика правового статуса судебного пристава в Российской империи и Российской Федерации (сравнительно-правовой аспект)

Based on the methodology of chrono-discrete monogeographic comparative law, we identify the comparative potential of bailiffs’ legal status in the Russian Empire and the Russian Federation. Until recently, the legislation of the Russian Federation did not give a clear idea of bailiffs’ legal position in the civil service. We believe that, speaking about a bailiff’s legal status and comparing the regulatory framework that regulates it, it is necessary to determine what elements constitute a bailiff’s legal status; to compare comparable, conditionally comparable and incomparable elements of this status. At present, this issue is more definite and developed in comparison with the pre-revolutionary period. Comparison of specific elements of legal status in different historical periods may lead to the development of prac-tical recommendations for improving the current legislation governing the structure and activities of the institution of bailiffs. We designate six ele-ments of bailiffs’ legal status, which are fundamentally comparable in rela-tion to the Russian Empire and the Russian Federation. We also note the need to take into account the specific historical situation in the studied periods, the specifics of the political, economic, and social structure, and the peculiarities of legal awareness and mentality.

The equivalence of uninorms induced by the U-partial order

In this paper, some properties of an order induced by uninorms are investigated. In this aim, the set of incomparable elements with respect to the $U$-partial order for any uninorm is introduced and studied. Also, by defining such an order, an equivalence relation on the class of uninorms is defined and this equivalence is deeply investigated. Finally, another set of incomparable elements with respect to the $U$-partial order for any uninorm is introduced and studied.

On shortening u-cycles and u-words for permutations

This paper initiates the study of shortening universal cycles (u-cycles) and universal words (u-words) for permutations either by using incomparable elements, or by using non-deterministic symbols. The latter approach is similar in nature to the recent relevant studies for the de Bruijn sequences. A particular result we obtain in this paper is that u-words for n-permutations exist of lengths n!+(1−k)(n−1) for k=0,1,…,(n−2)!.

Equivalence classes of uninorms

In this paper, some properties of an order induced by uninorms are studied. The set of incomparable elements with respect to the U-partial order for any uninorm on bounded lattices is investigated. Also, an equivalence relation on the class of uninorms induced by a U-partial order is investigated and discussed. Finally, the relationships between an order induced by uninorms and distributivity property for uninorms are investigated.

Ordered and delayed adversaries and how to work against them on a shared channel

An execution of a distributed algorithm is often seen as a game between the algorithm and a conceptual adversary causing specific distractions to the computation. In this work we define a class of ordered adaptive adversaries, which cause distractions—in particular crashes—online according to some partial order of the participating stations, which is fixed by the adversary before the execution. We distinguish: Linearly-Ordered adversary, restricted by some pre-defined linear order of (potentially) crashing stations; Anti-Chain-Ordered adversary, previously known as the Weakly-Adaptive adversary, which is restricted by some pre-defined set of crash-prone stations (it can be seen as an ordered adversary with the order being an anti-chain, i.e., a collection of incomparable elements, consisting of these stations); k-Thick-Ordered adversary restricted by partial orders of stations with a maximum anti-chain of size k. We initiate a study of how they affect performance of algorithms. For this purpose, we focus on the well-known Do-All problem of performing t tasks by p synchronous crash-prone stations communicating on a shared channel. The channel restricts communication by the fact that no message is delivered to the operational stations if more than one station transmits at the same time. The question addressed in this work is how the ordered adversaries controlling crashes of stations influence work performance, defined as the total number of available processor steps during the whole execution and introduced by Kanellakis and Shvartsman (Distrib Comput 5(4):201–217, 1992) in the context of Write-All algorithms. The first presented algorithm solves the Do-All problem with work {mathcal {O}}(t+p sqrt{t}log p) against the Linearly-Ordered adversary. Surprisingly, the upper bound on performance of this algorithm does not depend on the number of crashes f and is close to the absolute lower bound varOmega (t+psqrt{t}) proved in Chlebus et al. (Distrib Comput 18(6):435–451, 2006). Another algorithm is developed against the Weakly-Adaptive adversary. Work done by this algorithm is mathcal {O}(t + psqrt{t} + pmin left{ p/(p-f),tright} log p ), which is close to the lower bound varOmega (t + psqrt{t} + pmin left{ p/(p-f),tright} ) proved in [11] and answers the open questions posed there. We generalize this result to the class of k-Thick-Ordered adversaries, in which case the work of the algorithm is bounded by mathcal {O}(t + psqrt{t} + pmin left{ p/(p-f),k,tright} log p ). We complement this result by proving the almost matching lower bound Ω(t+pt+pminp/(p-f),k,t).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\varOmega (t + p\\sqrt{t} + p\\min \\left\\{ p/(p-f),k,t\\right\\} ). \\end{aligned}$$\\end{document}Independently from the results for the ordered adversaries, we consider a class of delayed adaptive adversaries, which could see random choices with some delay. We present an algorithm that works efficiently against the 1-RD adversary, which could see random choices of stations with one round delay, achieving close to optimal {mathcal {O}}(t+p sqrt{t}log ^{2} p) work complexity. This shows that restricting the adversary by not allowing it to react on random decisions immediately makes it significantly weaker, in the sense that there is an algorithm achieving (almost) optimal work performance.

Branch-and-Cut-and-Price algorithms for the preemptive RCPSP

In this article, we address the preemptive Resource-Constrained Precedence Scheduling Problem. We propose two mixed integer formulations containing an exponential number of variables and inequalities. An antichain is a set of pairwise incomparable elements with respect to the precedence constraints. In the first formulation, the integer variables are associated with the antichains. For the second, the integer variables are limited to a particular subset of antichains. We propose two Branch-and-Cut-and-Price algorithms for each of these formulations. We introduce some valid inequalities in order to reinforce the second formulation. Finally, we give some computational results on instances of the PSPLIB and compare the formulations.

On the properties of the F-partial order and the equivalence of nullnorms

In this study, we investigate a partial order induced by a nullnorm F on a bounded lattice L, called the F-partial order, which was recently introduced by Aşıcı. We consider the case where L=[0,1]. First, we define and comprehensively study the set of all incomparable elements with arbitrary but fixed x∈(0,1) according to the F-partial order. We then define and discuss the equivalence relation on the class of nullnorms according to the partial orders that they generate. By defining this relation, the equivalence relations defined on the class of t-norms and t-conorms are extended to a more general form. Finally, we provide an answer to an open problem regarding the relation between the lattice order and the F-partial order on [0,1].

Equivalence relations induced by the U-partial order

In this paper, an equivalence relation on the class of uninorms with the same neutral element induced by a U-partial order is provided and discussed. The equivalence classes linked to some special uninorms are characterized. Furthermore, the set of incomparable elements with respect to the U-partial order is deeply investigated. Finally, another equivalence relation on the class of uninorms with the same neutral element is introduced and studied.

THE LINEAR DISCREPANCY OF A PRODUCT OF TWO POSETS

For a poset <TEX>$P=(X,{\leq}_P)$</TEX>, the linear discrepancy of P is the minimum value of maximal differences of all incomparable elements for all possible labelings. In this paper, we find a lower bound and an upper bound of the linear discrepancy of a product of two posets. In order to give a lower bound, we use the known result, <TEX>$ld({\mathbf{m}}{\times}{\mathbf{n}})={\lceil}{\frac{mn}{2}}{\rceil}-2$</TEX>. Next, we use Dilworth's chain decomposition to obtain an upper bound of the linear discrepancy of a product of a poset and a chain. Finally, we give an example touching this upper bound.

An equivalence relation based on the U-partial order

In this paper, an equivalence relation on the class of uninorms induced by the U-partial order is discussed. Defining the set of all incomparable elements w.r.t. the U-partial order, this set is deeply investigated and some relations with the sets of all incomparable elements w.r.t. the orders induced by the corresponding underlying t-norm and t-conorm are presented. Also, the set of all incomparable elements with a fixed element w.r.t. the U-partial order is defined and studied in detail.

Well-quasi-ordering [formula omitted]-contraction-free graphs

A well-quasi-order is an order which contains no infinite decreasing sequence and no infinite collection of incomparable elements. In this paper, we consider graph classes defined by excluding one graph as contraction. More precisely, we give a complete characterization of graphs H such that the class of H-contraction-free graphs is well-quasi-ordered by the contraction relation. This result is the contraction analogue of the previous dichotomy theorems of Damsaschke (1990) on the induced subgraph relation, Ding (1992) on the subgraph relation, and Błasiok et al. (2015) on the induced minor relation.

An order induced by nullnorms and its properties

In this paper, an order induced by nullnorms on bounded lattices is given and discussed. We define the set of incomparable elements with respect to the order induced by a nullnorm. Also, by defining such an order, an equivalence relation on the class of nullnorms is defined and this equivalence is deeply investigated.