Abstract: The fundamental characterization theorem of standard elements in lattices is extended to posets. Several other characterizations of standard elements are obtained in a sectionally semi-complemented poset and also in an atomistic, dually sectionally semi-complemented poset.

Abstract: We prove that any countable (finite or infinite) partially ordered set may be represented by finite oriented paths ordered by the existence of homomorphism between them. This (what we believe a surprising result) solves several open problems. Such path-representations were previously known only for finite and infinite partial orders of dimension 2. Path-representation implies the universality of other classes of graphs (such as connected cubic planar graphs). It also implies that finite partially ordered sets are on-line representable by paths and their homomorphisms. This leads to new on-line dimensions.

Abstract: In this note we show how 1-factors in the middle two layers of the discrete cube can be used to construct 2-factors in the Odd graph (the Kneser graph of (k − 1)-sets from a (2k − 1)-set). In particular, we use the lexical matchings of Kierstead and Trotter, and the modular matchings of Duffus, Kierstead and Snevily, to give explicit constructions of two different 2-factorisations of the Odd graph.

Abstract: We study relations among the set of infinitesimal elements of pseudo MV-algebras and the problem of existence of states on them. This is important because in contrast to MV-algebras, it can happen that a pseudo MV-algebra has no states, so no probabilistic evaluation of events on it is possible. We introduce two kinds of radicals, and we deal with their relation. In some cases, they are completely different, which is not the case for MV-algebras. We give many interesting examples describing different situations, and we deal in more details with a subvariety of symmetric pseudo MV-algebras, where both complements coincide.

Abstract: We define and investigate the scale independent aggregation functions that are meaningful to aggregate finite ordinal numerical scales. Here scale independence means that the functions always have discrete representatives when the ordinal scales are considered as totally ordered finite sets. We also show that those scale independent functions identify with the so-called order invariant functions, which have been described recently. In particular, this identification allows us to justify the continuity property for certain order invariant functions in a natural way.

Abstract: A linear ordering is said to be representable if it can be order-embedded into the reals. Representable linear orderings have been characterized as those which are separable in the order topology and have at most countably many jumps. We use this characterization to study the repre- sentability of a lexicographic product of linear orderings. First we count the jumps in a lexicographic product in terms of the number of jumps in its factors. Then we relate the separability of a lexico- graphic product to properties of its factors, and derive a classification of representable lexicographic products. Mathematics Subject Classifications (2000): Primary 06A05; Secondary 06F30, 54F05, 91B16.

Abstract: Biorders, also called Ferrers relations, formalize Guttman scales. Irreflexive biorders on a set are exactly the interval orders on that set. The biorder polytope is the convex hull of the characteristic matrices of biorders. Its definition is thus similar to the definition of other order polytopes, the linear ordering polytope being the proeminent example. We investigate the combinatorial structure of the biorder polytope, thus obtaining a complete linear description in a specific case, and the automorphism group in all cases. Moreover, a class of facet-defining inequalities defined from weighted graphs is thoroughly analyzed. A weighted generalization of stability-critical graphs is presented, which leads to new facets even for the well-studied linear ordering polytope.

Abstract: We study abstract properties of intervals in the complete lattice of all κ-meet-closed subsets (κ-subsemilattices) of a κ-(meet-)semilattice S, where κ is an arbitrary cardinal number. Any interval of that kind is an extremally detachable closure system (that is, for each closed set A and each point x outside A, deleting x from the closure of A∪{x} leaves a closed set). Such closure systems have many pleasant geometric and lattice-theoretical properties; for example, they are always weakly atomic, lower locally Boolean and lower semimodular, and each member has a decomposition into completely join-irreducible elements. For intervals of κ-subsemilattices, we describe the covering relation, the coatoms, the ∨-irreducible and the ∨-prime elements in terms of the underlying κ-semilattices. Although such intervals may fail to be lower continuous, they are strongly coatomic if and only if every element has an irredundant (and even a least) join-decomposition. We also characterize those intervals which are Boolean, distributive (equivalently: modular), or semimodular.

Abstract: A (partially) ordered set P is well founded if no infinite decreasing sequences occur in P. A well founded poset containing no infinite antichains is called partially well ordered. We investigate some operations preserving that property and linear extensions of partial well orders.

Abstract: An ordered set-partition (or preferential arrangement) of n labeled elements represents a single “hierarchy” these are enumerated by the ordered Bell numbers. In this note we determine the number of “hierarchical orderings” or “societies”, where the n elements are first partitioned into m ≤ n subsets and a hierarchy is specified for each subset. We also consider the unlabeled case, where the ordered Bell numbers are replaced by the composition numbers. If there is only a single hierarchy, we show that the average rank of an element is asymptotic to n/(4 log 2) in the labeled case and to n/4 in the unlabeled case.

Abstract: This chapter is devoted to a discussion of quantum phase transitions in regularly alternating spin-1/2 Ising chain in a transverse field. After recalling some generally-known topics of the classical (temperature-driven) phase transition theory and some basic concepts of the quantum phase transition theory I pass to the statistical mechanics calculations for a one-dimensional spin-1/2 Ising model in a transverse field, which is the simplest possible system exhibiting the continuous quantum phase transition. The essential tool for these calculations is the Jordan-Wigner fermionization. The latter technique being completed by the continued fraction approach permits to obtain analytically the thermodynamic quantities for a `slightly complicated' model in which the intersite exchange interactions and on-site fields vary regularly along a chain. Rigorous analytical results for the ground-state and thermodynamic quantities, as well as exact numerical data for the spin correlations computed for long chains (up to a few thousand sites) demonstrate how the regularly alternating bonds/fields effect the quantum phase transition. I discuss in detail the case of period 2, swiftly sketch the case of period 3 and finally summarize emphasizing the effects of periodically modulated Hamiltonian parameters on quantum phase transitions in the transverse Ising chain and in some related models.

Abstract: Ohkuma [7] proved that if 〈L,<〉 is a totally ordered set with the property that its automorphism group is uniquely transitive (that is, for every a, b ∈ L there is a unique f ∈ Aut(〈L,<〉) such that f(a) = b), then 〈L,<〉 is isomorphic to an ordered subset 〈A,<〉 of the real numbers 〈R, <〉 such that 〈A,+〉 is also a subgroup of 〈R,+〉. Such ordered sets and groups have since been studied widely (see [3], for example) and called Ohkuma chains. Maroli [6] proved that even every nontrivial tree subject to the same unique transitivity condition is totally ordered, and thus is an Ohkuma chain. Other structures with the unique transitivity condition have been studied as well. For example, Giraudet and Holland [1] deal with uniquely transitive cyclically ordered sets, betweenness chains, and betweenness cyclically ordered sets, calling these Ohkuma structures. In that paper, it was shown that Ohkuma betweenness chains are closely related to chains with a slightly weaker transitivity condition, namely, that the group of order preserving automorphisms of these Ohkuma betweenness chains has two orbits, not just one. (See Theorem 14 of the present paper.)

Abstract: Commutator-finite D-lattices as a generalization of commutator-finite orthomodular lattices are defined and their properties studied. A necessary and sufficient condition is found under which a D-lattice can be uniquely decomposed into a direct product of an MV-algebra and finitely many irreducible D-lattices which are not MV-algebras. This condition is satisfied if the D-lattice is orthocomplete or if all commutators are sharp. A condition under which a block-finite D-lattice is commutator-finite is found. Some necessary and sufficient conditions for the existence of states and valuations are proved, and some examples are given.

Abstract: We define the (n,i,f)-tube orders, which include interval orders, trapezoid orders, triangle orders, weak orders, order dimension n, and interval-order-dimension n as special cases. We investigate some basic properties of (n,i,f)-tube orders, and begin classifying them by containment.

Abstract: We prove that a finite poset P=(V,≤) is determined up to some permutation of its elements by the function \(m_{P}\colon\ {V\choose2}\to\mathbf{N}_{0}\) defined by mP({u,v})=|{w∈V∣u≤w and v≤w}|.

Abstract: For a linearly ordered set (Z,≤) the length l(Z) of Z is the supremum of all cardinals that can be order-embedded or reverse order-embedded into Z. In this paper we give new proofs of two theorems relating the length and the cardinality of Z. The first one sets the following general inequality: |Z|≤2l(Z). The second one says that in the case that Z is a scattered chain (i.e. it does not contain rationals) we have |Z|=2l(Z).

Abstract: Let ℒ=〈L;∨,∧〉 be a subdirectly irreducible modular lattice, c∈L and p(x,y,z) an essentially ternary lattice term. In this paper we show that if p(x,y,c) is a semilattice operation then p(x,y,c)=∨ or ∧ and L is bounded and c=0 or c=1. This sheds light on the methodology used to move back and forth between generalizations of median algebras and lattices, and provides a negative answer to a problem posed by A. Knoebel and G. Meletiou.

Abstract: We present a characterisation of posets of size n with linear discrepancy n − 2. These are the posets that are “furthest” from a linear order without being an antichain.

Abstract: A structure is said to be ‘Okhuma’ if its automorphism group acts on it uniquely transitively, or slightly generalizing this, if its automorphism group acts uniquely transitively on each orbit. In this latter case we can think of the orbits as ‘colours’. Okhuma chains and related structures have been studied by Okhuma and others. Here we generalize their results to coloured chains, and give some constructions resulting from this of Okhuma graphs and digraphs.

Abstract: We introduce the class of bounded distributive lattices with two operators, Δ and ∇, the first between the lattice and the set of its ideals, and the second between the lattice and the set of its filters. The results presented can be understood as a generalization of the results obtained by S. Celani.

Abstract: The vertices of the flag graph Φ(P) of a graded poset P are its maximal chains. Two vertices are adjacent whenever two maximal chains differ in exactly one element. In this paper we characterize induced subgraphs of Cartesian product graphs and flag graphs of graded posets. The latter class of graphs lies between isometric and induced subgraphs of Cartesian products in the embedding structure theory. Both characterization use certain edge-labelings of graphs.

Abstract: We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W, S) and a subset K ⊆ S ,l etηK : wwK be the projection onto the parabolic subgroup WK . We show that the fibers of ηK constitute the smallest lattice congruence with 1 ≡ s for every s ∈ (S − K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order. Mathematics Subject Classifications (2000): Primary 20F55, 06B10; secondary 52C35.