Two easy duality theorems for product partial orders

AbstractTwo duality theorems are proved about the direct product of two partial orders. First, the size of the largest unichain (a chain fixed in one coordinate) equals the smallest number of semian-tichains (collections of elements in which no pair are comparable if they agree in either coordinate) needed to cover the elements of the product order. With analogous definitions, the size of a largest uniantichain equals the size of a smallest semichain covering

Minimizing the number of vehicles to meet a fixed periodic schedule : an application of periodic posets

Includes bibliographical references (leaf 25).Supported in part by the U.S. Army Research Office. DAAG29-80-C-0061by James B. Orlin

On calculating residuated approximations and the structure of finite lattices of small width

The concept of a residuated mapping relates to the concept of Galois connections; both arise in the theory of partially ordered sets. They have been applied in mathematical theories (e.g., category theory and formal concept analysis) and in theoretical computer science. The computation of residuated approximations between two lattices is influenced by lattice properties, e.g. distributivity. In previous work, it has been proven that, for any mapping f : L → [special characters omitted] between two complete lattices L and [special characters omitted], there exists a largest residuated mapping ρf dominated by f, and the notion of the shadow σ f of f is introduced. A complete lattice [special characters omitted] is completely distributive if, and only if, the shadow of any mapping f : L → [special characters omitted] from any complete lattice L to [special characters omitted] is residuated. Our objective herein is to study the characterization of the skeleton of a poset and to initiate the creation of a structure theory for finite lattices of small widths. We introduce the notion of the skeleton L˜ of a lattice L and apply it to find a more efficient algorithm to calculate the umbral number for any mapping from a ∼ finite lattice to a complete lattice. We take a maximal autonomous chain containing x as an equivalent class [x] of x. The lattice L˜ is based on the sets {[x] | x ∈ L}. The umbral number for any mapping f : L → [special characters omitted] between two complete lattices is related to the property of L˜. Let L be a lattice satisfying the condition that [x] is finite for all x ∈ L; such an L is called ∼ finite. We define Lo = {[special characters omitted][x] | x ∈ L} and fo = [special characters omitted]. The umbral number for any isotone mapping f is equal to the umbral number for fo, and [special characters omitted] for any ordinal number α. Let [special characters omitted] be the maximal umbral number for all isotone mappings f : L → [special characters omitted] between two complete lattices. If L is a ∼ finite lattice, then [special characters omitted]. The computation of [special characters omitted] is less than or equal to that of [special characters omitted], we have a more efficient method to calculate the umbral number [special characters omitted]. The previous results indicate that the umbral number [special characters omitted] determined by two lattices is determined by their structure, so we want to find out the structure of finite lattices of small widths. We completely determine the structure of lattices of width 2 and initiate a method to illuminate the structure of lattices of larger width

Lattices of choice functions and consensus problems

. In this paper we consider the three classes of choice functionssatisfying the three significant axioms called heredity (H), concordance (C) and outcast (O). We show that the set of choice functions satisfying any one of these axioms is a lattice, and we study the properties of these lattices. The lattice of choice functions satisfying (H) is distributive, whereas the lattice of choice functions verifying (C) is atomistic and lower bounded, and so has many properties. On the contrary, the lattice of choice functions satisfying(O) is not even ranked. Then using results of the axiomatic and metric latticial theories of consensus as well as the properties of our three lattices of choice functions, we get results to aggregate profiles of such choice functions into one (or several) collective choice function(s).Aggregation, choice function, concordance, consensus, distance, distributive, heredity, lattice, outcast

On optimal and near-optimal algorithms for some computational graph problems

PhD ThesisSome computational graph problems are considered in this thesis and algorithms for solving these problems are described in detail. The problems can be divided into three main classes, namely, problems involving partially ordered sets, finding cycles in graphs, and shortest path problems. Most of the algorithms are based on recursive procedures using depth-first search. The efficiency of each algorithm is derived and it can be concluded that the majority of the proposed algorithms are either optimal and near-optimal within a constant factor. The efficiency of the algorithms is measured by the time and space requirements for their implementation.Conselho Nacional de Pesquisas,Brazil: Universidade Federal do Rio de Janeiro, Brazil

Monotonic functions of finite posets

Classes of monotone functions from finite posets to chains are studied. These include order-preserving and strict order-preserving maps. When the maps are required to be bijective they are called linear extensions. Techniques for handling the first two types are closely related; whereas for linear extensions quite distinct methods are often necessary, which may yield results for order-preserving injections. First, many new fundamental properties and inequalities of a combinatorial nature are established for these maps. Quantities considered here include the range, height, depth and cardinalities of subposets. In particular we study convexity in posets and similarly pre-images of intervals in chains. The problem of extending a map defined on a subposet to the whole poset is discussed. We investigate positive correlation inequalities, having implications for the complexity of sorting algorithms. These express monotonicity properties for probabilities concerning sets of relations in posets. New proofs are given for existing inequalities and we obtain corresponding negative correlations, along with a generalization of the xyz inequality. The proofs involve inequalities in distributive lattices, some of which arose in physics. A characterization is given for posets satisfying necessary conditions for correlation properties under linear extensions. A log concavity type inequality is proved for the number of strict or non-strict order-preserving maps of an element. We define an explicit injection whereas the bijective case is proved in the literature using inequalities from the theory of mixed volumes. These results motivate a further group of such inequalities. But now we count numbers of strict or non-strict order-preserving maps of subposets to varying heights in the chain. Lastly we consider the computational cost of producing certain posets which can be associated with classical sorting and selection problems. A lower bound technique is derived for this complexity, incorporating either a new distributive lattice inequality, or the log concavity inequalities