Lexicographic sums and fibre-faithful maps

The notion of lexicographic sum is introduced in general categories. Existence criteria are derived, particularly for locally cartesian closed categories and for categories with suitable coproducts. Lexicographic sums satisfy a generalized associative law. More importantly, every morphism can be factored through the lexicographic sum of its fibres. This factorization and the two types of maps arising from it, fibre-trivial and fibre-faithful, are studied particularly for partially ordered sets and forT ₁-spaces.     

Weak relatively uniform convergences on abelian lattice ordered groups

The notion of relatively uniform convergence has been applied in the theory of vector lattices and in the theory of archimedean lattice ordered groups. Let G be an abelian lattice ordered group. In the present paper we introduce the notion of weak relatively uniform convergence (wru-convergence, for short) on G generated by a system M of regulators. If G is archimedean and M = G ⁺, then this type of convergence coincides with the relative uniform convergence on G. The relation of wru-convergence to the o-convergence is examined. If G has the diagonal property, then the system of all convex ℓ-subgroups of G closed with respect to wru-limits is a complete Brouwerian lattice. The Cauchy completeness with respect to wru-convergence is dealt with. Further, there is established that the system of all wru-convergences on an abelian divisible lattice ordered group G is a complete Brouwerian lattice.     

Distinguished extensions of a lattice-ordered group

Any lattice-ordered group (l-group for short) is essentially extended by its lexicographic product with a totally ordered group. That is, anl-homomorphism (i.e., a group and lattice homomorphism) on the extension which is injective on thel-group must be injective on the extension as well. Thus nol-group has a maximal essential extension in the categoryIGp ofl-groups withl-homomorphisms. However, anl-group is a distributive lattice, and so has a maximal essential extension in the categoryD of distributive lattices with lattice homomorphisms. Adistinguished extension of onel-group by another is one which is essential inD. We characterize such extensions, and show that everyl-groupG has a maximal distinguished extensionE(G) which is unique up to anl-isomorphism overG.E(G) contains most other known completions in whichG is order dense, and has mostl-group completeness properties as a result. Finally, we show that ifG is projectable then E(G) is the -completion of the projectable hull ofG.Presented by M. Henriksen.     

Cancellation and absorption of lexicographic powers of totally ordered Abelian groups

Let G and H be totally ordered Abelian groups such that, for some integer k, the lexicographic powers G ^k and H ^k are isomorphic (as ordered groups). It was proved by F. Oger that G and H need not be isomorphic. We show here that they are whenever G is either divisible or ₁ -saturated (and in a few more cases). Our proof relies on a general technique which we also use to prove that G and H must be elementary equivalent as ordered groups (a fact also proved by F. Delon and F. Lucas) and isomorphic as chains. The same technique applies to the question of whether G and H should be isomorphic as groups, but, in this last case, no hint about a possible negative answer seems available.     

Cantor extension of an abelian lattice ordered group equipped with a weak relatively uniform convergence

The notion of weak relatively uniform convergence (wru-convergence, for short) on an abelian lattice ordered group G has been investigated in a previous authors?? article. In the present paper we deal with Cantor extension of G and completion of G with respect to a wru-convergence on G.     

Relatively uniform convergences in archimedean lattice ordered groups

For an archimedean lattice ordered group G let G ^d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G ^d ∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

On Pseudocomplemented and Stone Ordered Sets

The aim of this paper is to characterize both the pseudocomplemented and Stone ordered sets in a manner similar to that used previously for Boolean and distributive ordered sets. The sublattice G(A) of the Dedekind–Mac Neille completion DM(A) of an ordered set A generated by A is said to be the characteristic lattice of A. We will show that there are distributive pseudocomplemented ordered sets whose characteristic lattices are not pseudocomplemented. We can define a stronger notion of pseudocomplementedness by demanding that both A and G(A) be pseudocomplemented. It turns out that the two concepts are the same for finite and Stone ordered sets.     

On <Emphasis Type="Italic">α</Emphasis>-complete retracts of specker lattice ordered groups

Let α be a cardinal. The notion of α-complete retract of a Boolean algebra has been studied by Dwinger. Specker lattice ordered groups were investigated by Conrad and Darnel. Assume that G is a Specker lattice ordered group generated by a Boolean algebra B(G). The notion of α-complete retract of G can be defined analogously as in the case of Boolean algebras. In the present paper we deal with the relations between α-complete retracts of G and α-complete retracts of B(G).     

Lexicographic product decompositions of half linearly ordered loops

In this paper we prove for an hl-loop Q an assertion analogous to the result of Jakubík concerning lexicographic products of half linearly ordered groups. We found conditions under which any two lexicographic product decompositions of an hl-loop Q with a finite number of lexicographic factors have isomorphic refinements.     

Distinguished Completion of a Direct Product of Lattice Ordered Groups

The distinguished completion E(G) of a lattice ordered group G was investigated by Ball [1], [2], [3]. An analogous notion for MV-algebras was dealt with by the author [7]. In the present paper we prove that if a lattice ordered group G is a direct product of lattice ordered groups G _i (i I), then E(G) is a direct product of the lattice ordered groups E(G _i). From this we obtain a generalization of a result of Ball [3].     

The Hahn–Banach Theorem for Partially Ordered Totally Convex, Positively Convex and Superconvex Modules

The Hahn–Banach Theorem for partially ordered totally convex modules [3] and a necessary and sufficient condition for the existence of an extension of a morphism from a submodule C ₀ of a partially ordered totally convex module C (with the ordered unit ball of the reals as codomain) to C, are proved. Moreover, the categories of partially ordered positively convex and superconvex modules are introduced and for both categories the Hahn–Banach Theorem is proved.     

Sequential convergences on cyclically ordered groups without Urysohn’s axiom

In this paper we investigate sequential convergences on a cyclically ordered group G which are compatible with the group operation and with the relation of cyclic order; we do not assume the validity of the Urysohn’s axiom. The system convG of convergences under consideration is partially ordered by means of the set-theoretical inclusion. We prove that convG is a Brouwerian lattice. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, grant I/2/2005.     

Real Closed Graded Fields

We investigate homogeneous orderings on G-graded rings where G is an arbitrary ordered abelian group. For this we introduce the notion of real closed graded fields. We generalize the Artin–Schreier characterization of real closed fields to the graded context. We also characterize real closed graded fields in terms of the group G and in terms of its homogeneous elements of degree 0. Supported by DFG-project KN202/5-1.     

On archimedean extensions ofAO-groups

The concept of almost orthogonality is a variant for extending the concept of the orthogonal elements of lattice-ordered groups to arbitrary partially ordered groups. In the present paper, the notion of an Archimedean extension of anAO-groups is studied. Some results are obtained concerning interpolationAO-groups. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.     

The geodetic number of the lexicographic product of graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S. The size of a minimum geodetic set in G is the geodetic number g(G) of G. We find that the geodetic number of the lexicographic product G°H for a non-complete graph H lies between 2 and 3g(G). We characterize the graphs G and H for which g(G°H)=2, as well as the lexicographic products T°H that enjoy g(T°H)=3g(G), when T is isomorphic to a tree. Using a new concept of the so-called geodominating triple of a graph G, a formula that expresses the exact geodetic number of G°H is established, where G is an arbitrary graph and H a non-complete graph.     

Orthocomplemented lattices in ordered groups

On a partially ordered set G the orthogonality relation is defined by incomparability and is a complete orthocomplemented lattice of double orthoclosed sets. We will prove that the atom space of the lattice has the same order structure as G. Thus if G is a partially ordered set (an ordered group, or an ordered vector space), then is a canonically partially ordered set (an ordered quotient group, or an ordered quotient vector space, respectively). We will also prove: if G is an ordered group with a positive cone P, then the lattice has the covering property iff , where g is an element of G and M is the intersection of all maximal subgroups contained in . Received August 1, 2006; accepted in final form May 29, 2007.     

Test sets for generalized matrix functions

Let G = (V,E) be a biconnected graph and let C be a cycle in G. The subgraphs of G identified with the biconnected components of the contraction of C in G are called the bridges of C. Associated with the set of bridges of a cycle C is an auxilliary graphical structure G_C called a bridge graph or an overlap graph. Such auxilliary graphs have provided important insights in classical graph theory, algorithmic graph theory, and complexity theory. In this paper, we use techniques from algorithmic combinatorics and complexity theory to derive canonical forms for cycles in bridge graphs. These canonical forms clarify the relationship between cycles in bridge graphs, the structure of the underlying graph G, and lexicographic order relations on the vertices of attachment of bridges of a cycle. The first canonical form deals with the structure of induced bridge graph cycles of length greater than three. Cycles of length three in bridge graphs are studied from a different point of view, namely that of the characterization of minimal elements in certain related posets: ordered bridge three-cycles (10 minimal elements), bridge three-cycles (5 minimal elements), bridge deletion three-cycles (infinite number, 7 classes), minor order (K ₅ K _3,3), chordal bridge three-cycles (13 minimal elements), contraction poset (5 minimal elements), cycle-minor poset (infinite number, 14 classes). These results, each giving a different insight into the structure of bridge three-cycles, follow as corollaries from the characterization of the 10 minimal elements of the ordered bridge three-cycle poset. This characterization is constructive and may be regarded as an extension of the classical Kuratowski's Theorem which follows as a corollary. Algorithms are described for constructing these various minimal elements in time O(∣E∣) or O(∣V∣) depending on the case. The first canonical form gives a constructive proof of the result that a graph is nonplanar if and only if it has a cycle C whose bridge graph G_C (alternatively, skew bridge graph) has a three-cycle. An algorithm is described that constructs this three-cycle in time O(∣E∣). This is best possible.     

Finitely Presented, Coherent, and Ultrasimplicial Ordered Abelian Groups

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only if G is finitely generated as a group, G⁺ is well-founded as a partially ordered set, and the set of minimal elements of G⁺\ {0} is finite. (ii) Torison-free, finitely presented partially ordered abelian groups can be represented as subgroups of some Zⁿ, with a finitely generated submonoid of (Z⁺)ⁿ as positive cone. (iii) Every unperforated, finitely presented partially ordered abelian group is Archimedean. Further, we establish connections with interpolation. In particular, we prove that a divisible dimension group G is a directed union of simplicial subgroups if and only if every finite subset of G is contained into a finitely presented ordered subgroup.     

New Ore‐Type Conditions for H‐Linked Graphs

For a fixed (multi)graph H, a graph G is H‐linked if any injection f: V(H)→V(G) can be extended to an H‐subdivision in G. The notion of an H ‐linked graph encompasses several familiar graph classes, including k‐linked, k‐ordered and k‐connected graphs. In this article, we give two sharp Ore‐type degree sum conditions that assure a graph G is H ‐linked for arbitrary H. These results extend and refine several previous results on H ‐linked, k‐linked, and k‐ordered graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:69–77, 2012