Intrinsic metric preserving maps on partially ordered groups

1996 ◽  
Vol 36 (1) ◽  
pp. 135-140 ◽  
Author(s):  
M. Jasem
Keyword(s):  
Intrinsic Metric ◽  
Partially Ordered ◽  
Ordered Groups

scholarly journals Partially ordered groups which act on oriented order trees

2010 ◽  
Vol 40 (5) ◽  
pp. 1527-1578
Author(s):  
Matthew Horak ◽  
Melanie Stein
Keyword(s):  
Partially Ordered ◽  
Ordered Groups

Compatible Tight Riesz Orders

1976 ◽  
Vol 28 (1) ◽  
pp. 186-200 ◽  
Author(s):  
A. M. W. Glass
Keyword(s):  
Geometric Characterization ◽  
Permutation Groups ◽  
Algebraic Characterization ◽  
Partially Ordered ◽  
Ordered Field ◽  
Ordered Groups

N. R. Reilly has obtained an algebraic characterization of the compatible tight Riesz orders that can be supported by certain partially ordered groups [13; 14]. The purpose of this paper is to give a “geometric“ characterization by the use of ordered permutation groups. Our restrictions on the partially ordered groups will likewise be geometric rather than algebraic. Davis and Bolz [3] have done some work on groups of all order-preserving permutations of a totally ordered field; from our more general theorems, we will be able to recapture their results.

scholarly journals Convergence in partially ordered groups

1973 ◽  
Vol 18 (3) ◽  
pp. 239-246
Author(s):  
Andrew Wirth
Keyword(s):  
Uniform Convergence ◽  
Banach Lattice ◽  
Order Convergence ◽  
Ordered Group ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Ordered Groups ◽  
Integrally Closed ◽  
Relative Uniform Convergence ◽  
New Type

AbstractRelative uniform limits need not be unique in a non-archimedean partially ordered group, and order convergence need not imply metric convergence in a Banach lattice. We define a new type of convergence on partially ordered groups (R-convergence), which implies both the previous ones, and does not have these defects. Further R-convergence is equivalent to relative uniform convergence on divisible directed integrally closed partially ordered groups, and to order convergence on fully ordered groups.

On Integration in Partially Ordered Groups

1983 ◽  
Vol 35 (2) ◽  
pp. 353-372 ◽  
Author(s):  
Panaiotis K. Pavlakos
Keyword(s):  
Vector Spaces ◽  
Order Structure ◽  
Topological Groups ◽  
Topological Vector Spaces ◽  
Lattice Group ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Measurable Functions ◽  
Ordered Vector Spaces ◽  
Definition Of

M. Sion and T. Traynor investigated ([15]-[17]), measures and integrals having values in topological groups or semigroups. Their definition of integrability was a modification of Phillips-Rickart bilinear vector integrals, in locally convex topological vector spaces.The purpose of this paper is to develop a good notion of an integration process in partially ordered groups, based on their order structure. The results obtained generalize some of the results of J. D. M. Wright ([19]-[22]) where the measurable functions are real-valued and the measures take values in partially ordered vector spaces.Let if be a σ-algebra of subsets of T, X a lattice group, Y, Z partially ordered groups and m : H → F a F-valued measure on H. By F(T, X), M(T, X), E(T, X) and S(T, X) are denoted the lattice group of functions with domain T and with range X, the lattice group of (H, m)-measurable functions of F(T, X) and the lattice group of (H, m)-elementary measurable functions of F(T, X) and the lattice group of (H, m)-simple measurable functions of F(T, X) respectively.

scholarly journals Cauchy completion of Abelian tight Riesz groups

1975 ◽  
Vol 19 (1) ◽  
pp. 62-73 ◽  
Author(s):  
B. F. Sherman
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Important Case ◽  
Product Theorem ◽  
Real Vector ◽  
Partially Ordered ◽  
Word Group ◽  
Central Product ◽  
Ordered Groups ◽  
Cauchy Completion

This paper concerns the completions of partially ordered groups introduced by Fuchs (1965a) and the author (to appear); the p.o. groups under consideration are, generally, abelian tight Riesz groups, and so, throughout, the word “group” will refer to an abelian group.In section 3 we meet the cornerstone of the work, the central product theorem, by means of which we can interpret the Cauchy completion of a tight Riesz group in terms of the completion of any of its o-ideals; one particularly important case arises when the group has a minimal o-ideal. Such a minimal o-ideal is o-simple, and in section 6 the completion of an isolated o-simple tight Riesz group is shown to be a tight Riesz real vector space.

scholarly journals Tight Riesz groups

1972 ◽  
Vol 13 (2) ◽  
pp. 224-240 ◽  
Author(s):  
R. J. Loy ◽  
J. B. Miller
Keyword(s):  
General Theory ◽  
Open Interval ◽  
Discrete Topology ◽  
Topological Groups ◽  
Ordered Group ◽  
Partially Ordered ◽  
Interval Topology ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Positive Elements

The theory of partially ordered topological groups has received little attention in the literature, despite the accessibility and importance in analysis of the group Rm. One obstacle in the way of a general theory seems to be, that a convenient association between the ordering and the topology suggests that the cone of all strictly positive elements be open, i.e. that the topology be at least as strong as the open-interval topology U; but if the ordering is a lattice ordering and not a full ordering then U itself is already discrete. So to obtain in this context something more interesting topologically than the discrete topology and orderwise than the full order, one must forego orderings which make lattice-ordered groups: in fact, the partially ordered group must be an antilattice, that is, must admit no nontrivial meets or joins (see § 2, 10°).

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