scholarly journals Cauchy completion of partially ordered groups

1974 ◽  
Vol 18 (2) ◽  
pp. 222-229 ◽  
Author(s):  
B. F. Sherman
Keyword(s):  
Partial Order ◽  
Open Interval ◽  
Macneille Completion ◽  
Partially Ordered ◽  
Interval Topology ◽  
Original Order ◽  
Ordered Groups ◽  
Cauchy Completion ◽  
Original Group ◽  
Topological Completion

A number of completions have been applied to p.o.-groups — the Dede kind-Macneille completion of archimedean l.o. groups; the lateral completion of l.o. groups (Conrad [2]); and the orthocompletion of l.o. groups (Bernau [1]). Fuchs in [3] has considered a completion of p.o. groups having a non-trivial open interval topology — the only l.o. groups of this form being fully ordered. He applies an ordering, which arises from the original partial order, to the group of round Cauchy filters over this topology; Kowaisky in [6] has shown that group, imbued with a suitable topology, is in fact the topological completion of the original group under its open interval topology. In this paper a slightly different ordering, also arising from the original order, is proposed for the group of round Cauchy filters; Fuchs' ordering can be obtained from this one as the associated order.

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°).

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 Some Krein-Milman theorems for order-convexity

1974 ◽  
Vol 18 (3) ◽  
pp. 257-261
Author(s):  
Andrew Wirth
Keyword(s):  
Large Class ◽  
Open Interval ◽  
Theoretic Result ◽  
Interval Topology ◽  
Original Order ◽  
Definition Of

Analogues of the Krein-Milman theorem for order-convexity have been studied by several authors. Franklin [2] has proved a set-theoretic result, while Baker [1] has proved the theorem for posets with the Frink interval topology. We prove two Krein-Milman results on a large class of posets, with the open-interval topology, one for the original order and one for the associated preorder. This class of posets includes all pogroups. Cellular-internity defined in Rn by Miller [3] leads to another notion of convexity, cell-convexity. We generalize the definition of cell-convexity to abelian l-groups and prove a Krein-Milman theorem in terms of it for divisible abelian l-groups.

Generalization of Hölder's Theorem to Ordered Modules

1969 ◽  
Vol 21 ◽  
pp. 149-157 ◽  
Author(s):  
T. M. Viswanathan
Keyword(s):  
Sufficient Conditions ◽  
Open Interval ◽  
Order Structure ◽  
Real Numbers ◽  
Ordered Group ◽  
Quotient Field ◽  
Interval Topology ◽  
Ordered Field ◽  
Topological Completion ◽  
Necessary And Sufficient

Hölder's theorem on archimedean groups states:An ordered (abelian) group G is order isomorphic to an ordered subgroup of the ordered group R of real numbers if and only if it is archimedean.We comprehend this theorem in the following setting: G is a Z-module and Ris the completion with respect to the open interval topology of the ordered field Q; Qitself is the ordered quotient field of the ordered domain Z.Rephrasing the situation, we raise the following question: We start with a fully ordered domain A,let Kbe its ordered quotient field. We endow Kwith the open interval topology and consider , the topological completion of K. Is it possible to impose a compatible order structure on and if this can be done, when can we say that an ordered A-module Mis order isomorphic to an ordered A-submodule of ? In Theorem 3.1, we obtain a set of necessary and sufficient conditions for this isomorphism to hold.

Representations of ordered groups with compatible tight Riesz orders

1975 ◽  
Vol 20 (3) ◽  
pp. 307-322 ◽  
Author(s):  
N. R. Reilly
Keyword(s):  
Interpolation Property ◽  
Open Interval ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Lattice Order ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Ordered Groups ◽  
Strengthened Form ◽  
The Given

A tight Riesz group G is a partially ordered group G that satisfies a strengthened form of the Riesz interpolation property. The term “tight” was introduced by Miller in (1970) and the tight interpolation property has been considered by Fuchs (1965), Miller (1973), (to appear), (preprint), Loy and Miller (1972) and Wirth (1973). If G is free of elements called pseudozeros then G is a non-discrete Hausdorff topological group with respect to the open interval topology U. Moreover the closure P of the cone P of the given order is the cone of an associated order on G. This allows an interesting interplay between the associated order, the tight Riesz order and the topology U. Loy and Miller found of particular interest the case in which the associated partial order is a lattice order. This situation was considered in reverse by Reilly (1973) and Wirth (1973), who investigated the circumstances under which a lattice ordered group, and indeed a partially ordered group, permits the existence of a tight Riesz order for which the initial order is the associated order. These tight Riesz orders were then called compatible tight Riesz orders. In Section one we relate these ideas to the topologies denned on partially ordered groups by means of topological identities, as described by Banaschewski (1957), and show that the topologies obtained from topological identities are precisely the open interval topologies from compatible tight Riesz orders.

The Completion of an Abelian l-Group

1975 ◽  
Vol 27 (5) ◽  
pp. 980-985 ◽  
Author(s):  
G. Otis Kenny
Keyword(s):  
Abelian Group ◽  
Open Interval ◽  
Partially Ordered ◽  
Interval Topology ◽  
Neighborhood Base

A directed partially ordered abelian group (G, ≦ ) is a tight Riesz group if for a1, a2, b1, b2 ∈ G with ai < bj, i, j = 1,2, there is an x ∈ G with ai < x < bj, i, j = 1, 2. The open interval topology on G is the topology having as a base the set of all open intervals (a, b) = {x ∈ G|a < x < b}. For any x ∈ G, a neighborhood base at x is the set of all open intervals (x — a, x + a) = x + ( — a, a) for a > 0.

Torsion classes of abelian cyclically ordered groups

2012 ◽  
Vol 62 (4) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Distributive Lattice ◽  
Partial Order ◽  
Torsion Class ◽  
Proper Class ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups

AbstractWe 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.

scholarly journals Primitive idempotent measures on compact semitopological semigroups

1972 ◽  
Vol 13 (4) ◽  
pp. 451-455 ◽  
Author(s):  
Stephen T. L. Choy
Keyword(s):  
Partial Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Zero Element ◽  
Primitive Idempotent ◽  
Natural Partial Order ◽  
Partially Ordered

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

Quotient groups and realization of tight Riesz groups

1973 ◽  
Vol 16 (4) ◽  
pp. 416-430 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Normal Subgroup ◽  
Open Interval ◽  
Canonical Homomorphism ◽  
Interval Topology ◽  
Essential Step ◽  
Image Position ◽  
Quotient Groups

Let (G, ≼) be an l-group having a compatible tight Riesz order ≦ with open-interval topology U, and H a normal subgroup. The first part of the paper concerns the question: Under what conditions on H is the structure of (G, ≼, ∧, ∨, ≦, U) carried over satisfactorily to by the canonical homomorphism; and its answer (Theorem 8°): H should be an l-ideal of (G, ≼) closed and not open in (G, U). Such a normal subgroup is here called a tangent. An essential step is to show that ≼′ is the associated order of ≦′.

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