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.     

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.     

Weak relatively uniform convergences on MV-algebras

Weak relatively uniform convergences (wru-convergences, for short) in lattice ordered groups have been investigated in previous authors’ papers. In the present article, the analogous notion for MV-algebras is studied. The system s(A) of all wru-convergences on an MV-algebra A is considered; this system is partially ordered in a natural way. Assuming that the MV-algebra A is divisible, we prove that s(A) is a Brouwerian lattice and that there exists an isomorphism of s(A) into the system s(G) of all wru-convergences on the lattice ordered group G corresponding to the MV-algebra A. Under the assumption that the MV-algebra A is archimedean and divisible, we investigate atoms and dual atoms in the system s(A).     

On a relative uniform completion of an archimedean lattice ordered group

The notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups. Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) of G is dealt with. It is known that exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in . Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by Grant VEGA 1/3003/06.     

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.     

Archimedean lattices

Anarchimedean lattice is a complete algebraic latticeL with the property that for each compact elementc∈L, the meet of all the maximal elements in the interval [0,c] is 0.L ishyper-archimedean if it is archimedean and for eachx∈L, [x, 1] is archimedean. The structure of these lattices is analysed from the point of view of their meet-irreducible elements. If the lattices are also Brouwer, then the existence of complements for the compact elements characterizes a particular class of hyper-archimedean lattices. The lattice ofl-ideals of an archimedean lattice ordered group is archimedean, and that of a hyper-archimedean lattice ordered group is hyper-archimedean. In the hyper-archimedean case those arising as lattices ofl-ideals are fully characterized.     

Groups acting on circulant matrices

Summary If a groupG permutes a setI, andM is a multiplicative abelian group, a representation ofG onM ^I is given by permutation of coordinates. TheG-module homomorphisms intoM ^I arise from exponential maps. This framework encompasses those systems of functional equations that characterize generalized hyperbolic functions.     

Torsion classes of abelian cyclically ordered groups

We introduce the notion of torsion class of abelian cyclically ordered groups; the definition is analogous to that used in the theory of lattice ordered groups. The collection T of all such classes is partially ordered by the class-theoretical inclusion. Though T is a proper class, we can apply the usual terminology for this partial order. We prove that T is a complete, infinitely distributive lattice having infinitely many atoms.     

Fields interpretable in rosy theories

We are working in a monster model ℭ of a rosy theory T. We prove the following theorems, generalizing the appropriate results from the finite Morley rank case and o-minimal structures. If R is a ⋁-definable integral domain of positive, finite U^t-rank, then its field of fractions is interpretable in ℭ. If A and M are infinite, definable, abelian groups such that A acts definably and faithfully on M as a group of automorphisms, M is A-minimal and U^t(M) is finite, then there is an infinite field interpretable in ℭ. If G is an infinite, solvable but non nilpotent-by-finite, definable group of finite U^t-rank and T has NIP, then there is an infinite field interpretable in 〈G, ·〉.     

p-harmonic 1-forms on complete manifolds

Let (M ^m , g) be a complete non-compact manifold with asymptotically non-negative Ricci curvature and finite first Betti number. We prove that any bounded set of p-harmonic 1-forms in L ^q (M), 0 < q < ∞, is relatively compact with respect to the uniform convergence topology.     

Relative uniform completeness and order-convex representations of archimedean ?-groups and f-rings

Results of Henriksen and Johnson, for archimedean f-rings with identity, and of Aron and Hager, for archimedean ?-groups with unit, relating uniform completeness to order-convexity of a representation in a D(X) (the lattice of almost real continuous functions on the space X) are extended to situations without identity or unit. For an archimedean ?-group, G, we show: if G admits any representation G?D(X) in which G is order-convex, then G is divisible and relatively uniformly complete. A converse to this would seem to require some sort of canonical representation of G, which seems not to exist in the ?-group case. But for a reduced archimedean f-ring, A, there is the Johnson representation A?D(XA), and we show: A is divisible, relatively uniformly complete and square-dominated if and only if A is order-convex in D(XA) and square-root-closed. Also, we expand on the situation with unit, where we have the Yosida representation, G?D(YG): if G is divisible, relatively uniformly complete, and the unit is a near unit, then G is order-convex in D(YG).     

A Distributive Lattice on the Set of Perfect Matchings of a Plane Bipartite Graph

Let G be a plane bipartite graph and M(G) the set of perfect matchings of G. The Z-transformation graph of G is defined as a graph on M(G): M,MM(G) are joined by an edge if and only if they differ only in one cycle that is the boundary of an inner face of G. A property that a certain orientation of the Z-transformation graph of G is acyclic implies a partially ordered relation on M(G). An equivalent definition of the poset M(G) is discussed in detail. If G is elementary, the following main results are obtained in this article: the poset M(G) is a finite distributive lattice, and its Hasse diagram is isomorphic to the Z-transformation digraph of G. Further, a distributive lattice structure is established on the set of perfect matchings of any plane bipartite graph.     

Multiplicative Jordan Decomposition in Group Rings of 3-Groups,II

In this paper, we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ?[G] satisfies the multiplicative Jordan decomposition (MJD). In the nonabelian case, we show that ?[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 3³ = 27.     

Many Familiar Categories can be Interpreted as Categories of Generalized Metric Spaces

The simple concepts of (general) distance function and homometry (a map that preserves distances up to a calibration) are introduced, and it is shown how some natural distance functions on various mathematical objects lead to concrete embeddings of the following categories into the resulting category DIST°: quasi-pseudo-metric, topological, and (quasi-)uniform spaces with various kinds of maps; groups and lattice-ordered abelian groups; rings and modules, particularly fields; sets with reflexive relations and relation-preserving maps (particularly directed loop-less graphs and quasi-ordered sets); measured spaces with Radon-continuous maps; Boolean, Brouwerian, and orthomodular lattices; categories with combined objects, for example topological groups, ordered topological spaces, ordered fields, Banach spaces with linear contractions or linear continuous maps and so on.     

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].     

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.     

Intrinsic metrics for lattice ordered groups

Swamy and Jakubik studied the metric ¦x y¦ on lattice ordered groups, and isometries which presere it. We show the only intrinsic metrics on lattice ordered groups are the multiplesn ¦x–y ¦ of theirs, and that the triangle inequality is satisfied by such a metric iff the group is abelian. We show that there are isometries for each of these metrics, but they are rare. We give a simpler proof via permutation groups of the following augmented version of a theorem of Jakubik. IfT is an isometry of the lattice ordered groupG with respect to the metric ¦x¥¦ andT(0)=0, thenG=AB, B is abelian, andT(a+b)=a–b; conversely, any suchT is an isometry.To Paul Conrad on his 60th birthdayPresented by L. Fuchs.     

Projective Resolutions for Modules over Infinite Groups

We define a notion of complexity for modules over group rings of infinite groups. This generalizes the notion of complexity for modules over group algebras of finite groups. We show that if M is a module over the group ring kG, where k is any ring and G is any group, and M has f-complexity (where f is some complexity function) over some set of finite index subgroups of G, then M has f-complexity over G (up to a direct summand). This generalizes the Alperin-Evens Theorem, which states that if the group G is finite then the complexity of M over G is the maximal complexity of M over an elementary abelian subgroup of G. We also show how we can use this generalization in order to construct projective resolutions for the integral special linear groups, SL(n, ℤ), where n ≥ 2.     

Invariant subspaces and cocycles in nonselfadjoint crossed products

Let G be a compact abelian group with the archimedean totally ordered dual Γ and let $\text{L}$ be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a one-parameter group {α_γ}_γ?Γ of trace preserving 1-automorphisms of M. In this paper, we investigate the structure of invariant subspaces and cocycles for the subalgebra $\text{L}$₊ of $\text{L}$ consisting of those operators whose spectrum with respect to the dual automorphism group {β_g}_g?G on $\text{L}$ is nonnegative. Our main result asserts that if M is a factor, then $\text{L}$₊ is maximal among the σ-weakly closed subalgebras of $\text{L}$.     

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).