Homomorphism Order Research Articles

On the Homomorphism Order of Labeled Posets

Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This universal order is a distributive lattice. We investigate some other properties, namely the infinite distributivity, the computation of infinite suprema and infima, and the complexity of certain decision problems involving the homomorphism order of k-posets. Sublattices are also examined.

On colorings of graph powers

In this paper, some results concerning the colorings of graph powers are presented. The notion of helical graphs is introduced. We show that such graphs are hom-universal with respect to high odd-girth graphs whose ( 2 t + 1 ) th power is bounded by a Kneser graph according to the homomorphism order. Also, we consider the problem of existence of homomorphism to odd cycles. We prove that such homomorphism to a ( 2 k + 1 ) -cycle exists if and only if the chromatic number of the ( 2 k + 1 ) th power of G 1 3 is less than or equal to 3, where G 1 3 is the 3-subdivision of G . We also consider Nešetřil’s Pentagon problem. This problem is about the existence of high girth cubic graphs which are not homomorphic to the cycle of size five. Several problems which are closely related to Nešetřil’s problem are introduced and their relations are presented.

Analogizing Hutton's quasi-uniformities for complete lattices and extending Shi's quasi-uniformities to closed set lattices

This paper gives analogies of Hutton's quasi-uniformities for complete lattices and extensions of Shi's pointwise quasi-uniformities to closed set lattices. CL GOH denotes the category of all complete lattices with generalized order homomorphisms as morphisms and CSL the category of all closed set lattices. HQUnif and SPQUnif denote the categories of Hutton quasi-uniform spaces on complete lattices and Shi pointwise quasi-uniform spaces on closed set lattices, respectively. We prove that both HQUnif and SPQUnif are fibre-complete topological categories over CL GOH and CSL with respect to the expected forgetful functors, respectively. Also, a mistake in W. Kotzé [Uniform spaces, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 553–580] is pointed out.

Generalised dualities and maximal finite antichains in the homomorphism order of relational structures

The motivation for this paper is threefold. First, we study the connectivity properties of the homomorphism order of directed graphs, and more generally for relational structures. As opposed to the homomorphism order of undirected graphs (which has no non-trivial finite maximal antichains), the order of directed graphs has finite maximal antichains of any size. In this paper, we characterise explicitly all maximal antichains in the homomorphism order of directed graphs. Quite surprisingly, these maximal antichains correspond to generalised dualities. The notion of generalised duality is defined here in full generality as an extension of the notion of finitary duality, investigated in [J. Nešetřil, C. Tardif, Duality theorems for finite structures (characterising gaps and good characterisations), J. Combin. Theory Ser. B 80 (1) (2000) 80–97]. Building upon the results of the cited paper, we fully characterise the generalised dualities. It appears that these dualities are determined by forbidding homomorphisms from a finite set of forests (rather than trees). Finally, in the spirit of [A. Atserias, On digraph coloring problems and treewidth duality, in: Proceedings of the 21st IEEE Symposium on Logic in Computer Science, LICS’06, IEEE Computer Society, 2006; B. Larose, C. Loten, C. Tardif, A characterisation of first-order constraint satisfaction problems, in: Proceedings of the 21st IEEE Symposium on Logic in Computer Science, LICS’06, IEEE Computer Society, 2006; V. Dalmau, A. Krokhin, B. Larose, First-order definable retraction problems for posets and reflexive graphs, in: Proceedings of the 19th IEEE Symposium on Logic in Computer Science, LICS’04, IEEE Computer Society, 2004 [5] ] we shall characterise “generalised” constraint satisfaction problems (defined also here) that are first-order definable. These are again just generalised dualities corresponding to finite maximal antichains in the homomorphism order.

On Finite Maximal Antichains in the Homomorphism Order

Abstract We show that for structures with at most two relations all finite maximal antichains in the homomorphism order correspond to finite homomorphism dualities. We also show that most finite maximal antichains in this order split.

On the Density of Trigraph Homomorphisms

An order is dense if A < B implies A < C < B for some C. The homomorphism order of (nontrivial) graphs is known to be dense. Homomorphisms of trigraphs extend homomorphisms of graphs, and model many partitions of interest in the study of perfect graphs. We address the question of density of the homomorphism order for trigraphs. It turns out that there are gaps in the order, and we exactly characterize where they occur.

SINGLE ELEMENTS IN SOME REFLEXIVE ALGEBRA MODULES

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in Alg$\mathcal{L}$-module $\mathcal{U}_{\phi}$ is initiated, where $\mathcal{L}$ is a completely distributive subspace lattice on a Banach space $\mathcal{X}$ and $\phi$ is an order homomorphism from $\mathcal{L}$ into $\mathcal{L}$. For a reflexive Banach space $\mathcal{X}$ and a positive integer $n$ (or $+\infty$), by virtue of the order homomorphism $\phi$ we give necessary and sufficient conditions for the existence of single elements of $\mathcal{U}_{\phi}$ of rank $n$ (or $+\infty$).

SINGLE ELEMENTS IN SOME REFLEXIVE ALGEBRA MODULES

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in AlgL-module Uφ is initiated, where L is a completely distributive subspace lattice on a Banach space X and φ is an order homomorphism from L into L. For a reflexive Banach space X and a positive integer n (or +∞), by virtue of the order homomorphism φ we give necessary and sufficient conditions for the existence of single elements of Uφ of rank n (or +∞).

Minimal universal and dense minor closed classes

We study the homomorphism (coloring) order induced on minor closed classes. In [J. Hubička, J. Nešetřil, Finite Paths are Universal, ITI Series 2003-129, Charles University, 2003. Order (in press)], the minor closed class P of directed paths is shown to be universal and in [J. Nešetřil, X. Zhu, Path homomorphisms, Proc. Comb. Phil. Soc. (1996) 207–220], P is shown to contain a dense subset. In this note we prove that P is a unique minimal class of oriented graphs which is both universal and dense. Moreover, we show a dichotomy result for any minor closed class K of directed trees: K is either universal or it is well-quasi-ordered (wqo). We also prove structure theorems about series–parallel graphs (SPG), in an attempt to determine the minimal universal and dense minor closed classes of undirected graphs. We show the non-existence of universal classes in certain subclasses of SPG. Also for basic graphs in the class of SPG, we show that there is a linear time algorithm that decides whether such a graph is core or not. We also give a constructive description of arbitrary 2-connected graphs in SPG.

On a new reformulation of Hadwiger's conjecture

Assuming that every proper minor closed class of graphs contains a maximum with respect to the homomorphism order, we prove that such a maximum must be homomorphically equivalent to a complete graph. This proves that Hadwiger's conjecture is equivalent to saying that every minor closed class of graphs contains a maximum with respect to homomorphism order. Let F be a finite set of 2-connected graphs, and let C be the class of graphs with no minor from F . We prove that if C has a maximum, then any maximum of C must be homomorphically equivalent to a complete graph. This is a special case of a conjecture of Nešetřil and Ossona de Mendez.

Folding

We define folding of a directed graph as a coloring (or a homomorphism) which is injective on all the down sets of a given depth. While in general foldings are as complicated as homomorphisms for some classes they present useful tool to study colorings and homomorphisms. Our main result yields for any proper minor closed class K a folding (of any prescribed depth) using a fixed number of colors. This in turn yields (for any K ) the existence of a K k -free graph which bounds (in the homomorphism order) all K k -free graphs belonging to K . This has been conjectured in [J. Nešetřil, Aspects of structural combinatorics, Taiwanese J. Math. 3 (4) (1999) 381–424] and elsewhere and solved for k = 3 in [J. Nešetřil, P. Ossona de Mendez, Colorings and homomorphisms of minor closed classes, in: J. Pach, et al. (Eds.), J. Goodman and R. Pollack Festschrift, Springer, Berlin, 2003, pp. 651–664]. Particularly, we prove (without using 4CT) the existence of a graph H which satisfies χ ( H ) ⩽ 5 , ω ( H ) ⩽ 4 and such that any planar graph G is homomorphic to H. This is sandwiched between 4CT and 5CT for planar graphs and the general case has bearing to Hadwiger conjecture.

RANK-ONE OPERATORS IN REFLEXIVE A-SUBMODULES OF OPERATOR ALGEBRAS

In this paper, we first show that any reflexive $\mathcal{A}$-submodule $\mathcal{U}$ of a unital operator algebra $\mathcal{A}$ in $\mathcal{B(H)}$ is precisely of the following form: $$\mathcal{U} = \{ T \in \mathcal{B(H)}: TE \subseteq \phi(E) \quad \forall E \in \mathrm{Lat}\mathcal{A} \},$$ where $\phi$ is an order homomorphism of $\mathrm{Lat}\mathcal{A}$ into itself. Furthermore we investigate the density of the rank-one submodule of a reflexive $\mathcal{A}$-submodule in the $w^{*}$-topology and in certain pointwise approximation, and obtain several equivalent conditions by means of the order homomorphism $\phi$.

RANK-ONE OPERATORS IN REFLEXIVE A-SUBMODULES OF OPERATOR ALGEBRAS

In this paper, we first show that any reflexive $\cal A$-submodule $\cal U$ of a unital operator algebra $\cal A$ in $\cal B(H)$ is precisely of the following form: $$\cal U=\{T\in\cal B(H): TE\subseteq\phi(E) \quad\forall E\in\mbox{Lat}\cal A\},$$ where $\phi$ is an order homomorphism of $\mbox{Lat}\cal A$ into itself. Furthermore we investigate the density of the rank-one submodule of a reflexive $\cal A$-submodule in the $w^{*}$-topology and in certain pointwise approximation, and obtain several equivalent conditions by means of the order homomorphism $\phi$.

Tensor products of $\sigma $-weakly closed nest algebra submodules

In this paper we prove that for any unital cr-weakly closed algebra A which is σ-weakly generated by finite-rank operators in A, every σ-weakly closed A-submodule has Property S σ . In the case of nest algebras, if L 1 ,..., L n are nests, we obtain the following n-fold tensor product formula: u Φ1 ⊗...⊗u Φn = u Φ1⊗...⊗Φn , where each u Φi is the σ-weakly closed AlgL i -submodule determined by an order homomorphism Φ i from L i into itself.

The Morality of Unenabled Embryo Use—Arguments That Work and Arguments That Don't

The Morality of Unenabled Embryo Use—Arguments That Work and Arguments That Don't

Generalization of L-closure spaces

This paper introduces the concepts of L-closure spaces and the convergence in L-closure spaces. In the variable-basis setting of such closure spaces, continuous generalized order homomorphisms are characterized by the convergence theory of nets, filters and ideals. Furthermore, in the fixed-basis setting of such closure spaces, some constructions on L-closure spaces, such as the subspace of an L-cs, the sum L-cs, the product of L-cs’ and the induced L-cs of a crisp closure space, are investigated. We obtain an important result that the category L- CLOSURE is a topological category over SET w.r.t. the forgetful functor from L- CLOSURE to SET. We obtain another important result that there is an adjunction between the category of crisp closure spaces and the full subcategory of those L-closure spaces which are induced by some crisp closure space.

On maximal finite antichains in the homomorphism order of directed graphs

We show that the pairs {T, DT } where T is a tree and DT its dual are the only maximal antichains of size 2 in the category of directed graphs endowed with its natural homomorphism ordering.

Completeness and an extension theorem for continuous fuzzy linear order-homomorphisms

In this paper, the notions of λ-Cauchy net and completeness in L-topological vector spaces are introduced, and some of their properties are studied. An extension theorem for continuous fuzzy linear order homomorphisms is proved.

Interpolation in nest algebra modules

Let A be a nest algebra and LatA its invariant projection (or subspace) lattice. In this paper, using order homomorphisms of LatA, we give necessary and sufficient conditions on bounded linear operators X and Y on a Hilbert space to guarantee the existence of an operator A in a certain A-module such that AX = Y.

THE SET THEORETIC AMBIT OF ARROW'S THEOREM

Set theoretic formulation of Arrow's theorem, viewed in light of a taxonomy of transitive relations, serves to unmask the theorem's understated generality. Under the impress of the independence of irrelevant alternatives, the antipode of ceteris paribus reasoning, a purported compiler function either breaches some other rationality premise or produces the effet Condorcet. Types of cycles, each the seeming handiwork of a virtual voter disdaining transitivity, are rigorously defined. Arrow's theorem erects a dilemma between cyclic indecision and dictatorship. Maneuvers responsive thereto are explicable in set theoretic terms. None of these gambits rival in simplicity the unassisted escape of strict linear orderings, which, by virtue of the Arrow-Sen reflexivity premise, are not captured by the theorem. Yet these are the relations among whose n-tuples the effet Condorcet is most frequent. A generalization and stronger theorem encompasses these and all other linear orderings and total tierings. Revisions to the Arrow-Sen definitions of 'choice set' and 'rationalization' similarly enable one to generalize Sen's demonstration that some rational choice function always exists. Similarly may one generalize Debreu's theorems establishing conditions under which a binary relation may be represented by a continuous real-valued order homomorphism. Arrow has recounted that an early interest in mathematical logic, piqued by the work of Russell and a course from Tarksi, formed the seedbed for his celebrated impossibility theorem on the 'rational' aggregation of certain binary relations. 1 Arrow's departure from the nomenclature of set theory to some extent clouds the set theoretic compass of the theorem. This point has itself been eclipsed in a rich discussion of related impossibility results and of consequences for economic decisionmaking. The following treatment first casts the theorem and the occurrence of cycles in precise set theoretical terms. It then traces in the same terms responses to the dilemma that the theorem poses. The fruit is the discovery that the theorem is at once uneconomical and unnecessarily restricted in a respect whose cure furnishes a stronger result and generalization. The same is also true of Sen's companion results on the inferability of certain choice functions. These generalizations may seem within economics to enlarge scope only by including relations infrequently exhibited by consumers. Their general theoretical significance lies in the sweep of conclusions across all manner of transitive relations that position each member of a set with respect to each other.