We consider the poset of all posets on n elements where the partial order is that of inclusion of comparabilities. We discuss some properties of this poset concerning its height, width, jump number and dimension. We also give algorithms to construct some maximal chains in this poset which have special properties for these parameters.

Voronoi diagrams and Delaunay triangulations have been extensively used to represent and compute geometric features of point configurations. We introduce a generalization to poset diagrams and poset complexes, which contain order-k and degree-k Voronoi diagrams and their duals as special cases. Extending a result of Aurenhammer from 1990, we show how to construct poset diagrams as weighted Voro...

In this paper we describe a novel a procedure to build a linear order from an arbitrary poset which (i) preserves the original ordering and (ii) allows to extend monotonic and antitonic mappings defined over the original poset to monotonic and antitonic mappings over the new linear poset.

We study different problems of enumeration of standard paths in the poset of compositions of integers. We show that several problems similar to those considered in the poset of partitions of integers become simpler in this context. We give explicit formulas for generating functions of standard paths in this poset and interesting subposets, and a closed formula for the number of standard paths e...

The standard poset transit function of a poset P is a function TP that assigns to a pair of comparable elements the interval between them, while TP (x, y) = {x, y} for a pair x, y of incomparable elements. Posets in which the standard poset transit function coincides with the shortest-path transit function of its cover-incomparability graph are characterized in three ways, in particular with fo...

Term occurrences of any clause C are determined by their positions. The set of all term partitions defined on subsets of term occurrences of C form a partially ordered set. This poset is isomorphic to the set of all generalizations of C. The structure of this poset can be inferred from the term occurrences in C alone. We can apply these constructions in this poset in machine learning.

A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset, determine whether there exists polytope whose face lattice poset. In 1990s, Kapranov defined poset as hybrid between of permutohedron that an associahedron, he asked this realizable. Shortly after his question was posed, Reiner Ziegler provided realization. Based on our previous work ne...

In this paper X denotes a set. Let L be a lattice. Note that Poset(L) has l.u.b.’s and g.l.b.’s. Let L be an upper-bounded lattice. One can verify that Poset(L) is upper-bounded. Let L be a lower-bounded lattice. One can verify that Poset(L) is lower-bounded. Let L be a complete lattice. Note that Poset(L) is complete. Let X be a set. Then X is an order in X . Let X be a set. The functor 〈X ,⊆〉...

We characterize the finite distributive lattices which admit a complete valuation, that is bijective over a set of consecutive natural numbers, with the additional conditions of completeness (Definition 2.3). We prove that such lattices are downset lattices of finite posets of dimension at most two, and determine a realizer through a recursive relation between weights on the poset associated to...

Neural networks and fuzzy systems have been the subject of much interest in recent years for the control of nonlinear processes. Much research has been directed at indirect control schemes where, basically, the neural network or fuzzy system has been used to identify the process, and a controller has been synthesised from this model. An alternative approach is that of direct control where the n...