Union Of Chain Research Articles

Categoricity in multiuniversal classes

The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of multiuniversal classes. Roughly speaking, these are classes with a closure operator that is essentially algebraic closure (instead of, in the universal case, being essentially definable closure). Along the way, we prove in particular that Galois (orbital) types in multiuniversal classes are determined by their finite restrictions, generalizing a result of the second author.

A linear algorithm for the shortest transformation of graphs with different operation costs

A novel time- and memory-efficient algorithm for solving the problem of finding the most economical (i.e., having the lowest overall cost) transformation of an arbitrary oriented graph representing a disjoint union of chains and cycles into another graph of the same type is proposed. The correctness of this algorithm (i.e., the fact that it always yields the minimum of the overall cost functional) and the linearity of the estimated memory and time of its operation are demonstrated.

Homomorphisms and chains of Kripke models

In this paper we define a suitable version of the notion of homomorphism for Kripke models of intuitionistic first-order logic and characterize theories that are preserved under images and also those that are preserved under inverse images of homomorphisms. Moreover, we define a notion of union of chain for Kripke models and define a class of formulas that is preserved in unions of chains. We also define similar classes of formulas and investigate their behavior in Kripke models. An application to intuitionistic first-order arithmetic is also given.

Ideal chains in radical theory

It is shown that in a well known characterization of radical classes, closure under unions of chains of ideals can be replaced by closure under unions of continuous well-ordered chains of ideals. Some consequences are discussed.

Unions of slender groups

It is well-known that the class of slender groups is closed under extensions, arbitrary direct sums and subgroups. We show that it is also closed under taking unions of continuous well-ordered ascending chains of groups where each factor of consecutive groups is slender (Theorem 2.2). We also prove that the class of slender groups cannot be obtained from any set of slender groups by forming unions of chains of the mentioned kind and taking subgroups (Theorem 4.2). This generalizes a result by Gobel-Wald [11].

The Chang-Łoś-Suszko theorem in a topological setting

The Chang-Łoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications.

On the number of independent sets in damaged Cayley graphs

The Cayley graph generated by a set A is the graph ΓA(V) on a set of positive integers V such that a pair (u, v) ∈ V×V is an edge of the graph if and only if |u−v| ∈ A or u+v ∈ A. We denote by (n,m) the class of graphs G = (V, E) such that G is a union of chains and cycles and |V| = n, |E| = m. In this paper, we present an upper bound for the number of independent sets in Cayley graphs ΓA(V) such that A = {⌈n/2⌉ − t, ⌈n/2⌉ − ƒ}, V ⊆ [⌊n/2⌋ + 1, ⌊n/2⌋+t] ∪ [n − t + 1, n], where n, t, ƒ ∈ N and ƒ < t < n/4. We also describe the graph with the maximum number of independent sets in the family (n, m).

Unions of chains in dyadic compact spaces and topological groups

The following problem is considered: If a topological group G is the union of an increasing chain of subspaces and certain cardinal invariants of the subspaces are known, what can be said about G? We prove that if G is locally compact and every subspace in the chain has countable pseudocharacter or tightness, then G is metrizable. We also prove a similar assertion for σ-compact and totally bounded groups represented as the union of first countable subspaces, when the length of the chain is a regular cardinal greater than ω 1. Finally, we show that these results are not valid in general, not even for compact spaces.

A Criterion for Complete Decomposability and Butler Modules over Valuation Domains

Throughout this paper R will denote a valuation domain. P. Eklof and w x L. Fuchs 3 showed that an R-module M is free if and only M is a Baer 1 Ž . module, that is, Ext M, T s 0 for all torsion R-modules T. What can we R 1 Ž . say about R-modules M for which Bext M, T s 0 for all torsion modR ules T ? Here Bext is a subfunctor of Ext consisting of balanced R R extensions. Torsion-free R-modules M with the last property are called Butler modules. Is a Butler R-module M completely decomposable? L. w x Fuchs and E. Monari-Martinez 7 answered this in the affirmative when rank M F / . This paper attempts to investigate this question using the 1 w x recent work 11 on modules with balanced-projective dimension one. Ž . It is known that every torsion-free R-module M possesses a G / 9-family 0 Ž . C of pure submodules, namely, C satisfies the following conditions: 1 0, Ž . Ž . M g C , 2 C is closed under unions of chains, and 3 for any A g C and any countable subset X of M, there is a B g C such that A j X ; B and BrA has countable rank. For instance, we can take C to consist of all the Ž . pure submodules of M. If the condition 2 above is replaced by the Ž . condition 2 9 S N g C if N g C for each i g I, then we call C an ig I i i axiom-39 family. These are modifications of the original definition of an Ž w x. axiom-3 family by Paul Hill see, e.g., 1 and have also occurred with w x other variations under the name tight systems in 8 . Ž . Our main theorem Theorem 4.6 asserts that a Butler R-module B will be completely decomposable if and only if B has an axiom-39 family of pure submodules. We first show that the above condition on B is equiva-

Nonnegative jordan bases

In this paper we are concerned with the problem of when the generalized eigenspace of an n × n nonegative matrix A corresponding to its spectrai radius ρ has a nonnegative Jordan basis. Richman and Schneider have shown this to hold when, and only when, the Weyr characteristics of A corresponding to ρ is equal to the vector of levels in the singular graph of A Here we develop sufficient conditions for this equality to hold based upon the linear independence of a certain set of vectors which are generated according to a process of elimination of totally independent chains from the singular graph. We show that this condition comes close to being a further characterization for the existence of a nonnegative Jordan basis for this eigenspace and actually conjecture that this is the case. Finally, we show that this eigenspace has a Rothblum basis which is a Jordan basis if and only if the singular graph is a disjoint union of chains.

Saturated structures, unions of chains, and preservation theorems

Saturated structures, unions of chains, and preservation theorems

On classes closed under unions of chains

AbstractWe improve a general theorem of J. A. Makowsky which characterizes, for a wide class of languages, those sentences θ such that both Mod(θ) and Mod(¬θ) are closed under unions of chains.

Generalization of Dilworth's Theorem on Minimal Chain Decomposition

The decomposition of a finite partially ordered set of elements as a union of chains was considered by Dilworth [2]. Dantzig and Hoffman [1] formulated this problem as a linear programming problem and obtained Dilworth's theorem from duality theory. For some practical applications and for a method to obtain a minimal decomposition see Ford and Fulkerson [3]. In this paper we generalize this problem to the case when the set is not necessarily partially ordered and obtain a method of finding a minimal chain decomposition of the set.

On Unions of Chains of Models

On Unions of Chains of Models