Disjoint subsets of a partially ordered group

1969 ◽  
Vol 20 (6) ◽  
pp. 572-577
Author(s):  
J. Jakubík
Keyword(s):  
Ordered Group ◽  
Partially Ordered ◽  
Partially Ordered Group

Convex Directed Subgroups of a Group of Divisibility

1974 ◽  
Vol 26 (3) ◽  
pp. 532-542 ◽  
Author(s):  
Joe L. Mott
Keyword(s):  
Integral Domain ◽  
Multiplicative Group ◽  
Ordered Group ◽  
Quotient Field ◽  
Partially Ordered ◽  
Group Of Units ◽  
Partially Ordered Group ◽  
Group Of Divisibility

If D is an integral domain with quotient field K, the group of divisibility G(D) of D is the partially ordered group of non-zero principal fractional ideals with aD ≦ bD if and only if aD contains bD. If K* denotes the multiplicative group of K and U(D) the group of units of D, then G(D) is order isomorphic to K*/U(D), where aU(D) ≦ bU(D) if and only if b/a ∊ D.

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.

Generalized intervals in partially ordered groups

1959 ◽  
Vol 55 (2) ◽  
pp. 165-171 ◽  
Author(s):  
D. C. J. Burgess
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Ordered Group ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Ordered Groups ◽  
Normal Sense

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.

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.

scholarly journals Partially Ordered Group of the 2×2 Symmetric Matrices

2019 ◽  
Vol 1 (2) ◽  
Author(s):  
Irmatul Hasanah
Keyword(s):  
Symmetric Matrix ◽  
Positive Cone ◽  
Matrix Group ◽  
Symmetric Matrices ◽  
Ordered Group ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
The Matrix

AbstractThis paper deals with the partially ordered group of the 2×2 symmetric matrices. A matrix is defined to be a positive if each entry of the matrix is positive. With the characterization of the 2×2 symmetric matrix, we construct the positive cone such that the set of the matrices constructs partially ordered group.Keywords: Symmetric Matrix, Group-Ordered, Positive Cone. AbstrakArtikel ini membahas grup terurut parsial pada matriks simetri berukuran 2×2. Suatu matriks dikatakan positif jika setiap entri pada matriks bernilai positif. Melalui karakteristik dari matriks simetri, akan dikonstruksi sebuah positif cone sehingga matriks simetri berukuran 2×2  merupakan grup terurut parsial.Kata kunci: Matriks Simetri, Grup Terurut, Positif Cone.

Semi-Valuations and Groups of Divisibility

1969 ◽  
Vol 21 ◽  
pp. 576-591 ◽  
Author(s):  
Jack Ohm
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Ordered Group ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Theoretic Problem ◽  
Group Of Divisibility

Associated with any integral domain R there is a partially ordered group A, called the group of divisibility of R. When R is a valuation ring, A is merely the value group; and in this case, ideal-theoretic properties of R are easily derived from corresponding properties of A, and conversely. Even in the general case, though, it has proved useful on occasion to phrase a ring-theoretic problem in terms of the ordered group A, first solve the problem there, and then pull back the solution if possible to R. Lorenzen (15) originally applied this technique to solve a problem of Krull, and Nakayama (16) used it to produce a counterexample to another question of Krull. More recently, Heinzer (7;8) has used the method to construct other interesting examples of rings.

scholarly journals Compatible tight Riesz orders and prime subgroups

1973 ◽  
Vol 14 (2) ◽  
pp. 145-160 ◽  
Author(s):  
N. R. Reilly
Keyword(s):  
Interpolation Property ◽  
Finite Width ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Lattice Order ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Strengthened Form

A tight Riesz group is a partially ordered group which satisfies a strengthened form of the Riesz interpolation property. The term “tight” was introduced by Miller in [8], and the tight interpolation property has been considered in papers by Fuchs [3], Miller [8, 9], Loy and Miller [7] and Wirth [12]. If the closure of the cone P, in the interval topology, of such a partially ordered group G contains no pseudozeros, then is itself the cone of a partial order on G. Loy and Miller found of particular interest the case in which this associated partial orderis a lattice order. This situation was then considered in reverse by A. Wirth [12] who investigated under what circumstances a lattice ordered group would permit the existence of a tight Riesz order (called a compatible tight Riesz order) for which the initial lattice order is the order defined by the closure of the cone of the tight Riesz order.Wirth gave two fundamental anduseful characterizations of those subsets of the cone of a lattice ordered group that canbe the strict cone of a compatible tight Riesz order; one is in terms of archimedean classes and the other is an elementwise characterization. Although Loy, Miller and Wirth restricted their attention to abelian groups, much of what they do carries over verbatim to nonabelian groups. In the main result of this paper (Theorem 2.6) a description of the strict cone of a compatible tight Riesz order on a lattice ordered group Gis given in terms of the prime subgroups of G.This is particularly useful when one is attempting to identify the compatible tight Riesz orders on some particular lattice ordered group or class of lattice ordered groups, since it narrows down to a convenient family of subsets the possible candidates for strict cones of compatible tight Riesz orders. These can then be tested under Wirth's criteria. This technique is illustrated in § 5, where the compatible tight Riesz orders are determined o a lattice ordered group of the type V(Γ, Gγ), where Γ is of finite width, and in § 6, where two examples are considered.

Clean unital ℓ-groups

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
Anthony Hager ◽  
Chawne Kimber ◽  
Warren McGovern
Keyword(s):  
Lattice Ordered Group ◽  
Order Unit ◽  
Ordered Group ◽  
Partially Ordered ◽  
Clean Rings ◽  
The Past ◽  
Partially Ordered Group ◽  
Ordered Groups ◽  
Relationship Of ◽  
The Relationship

AbstractA ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be a notion of a clean partially-ordered group. In this article we define a clean unital lattice-ordered group; we state and prove a theorem which characterizes clean unital ℓ-groups. We mention the relationship of clean unital ℓ-groups to algebraic K-theory. In the last section of the article we generalize the notion of clean to the non-unital context and investigate this concept within the framework of W-objects, that is, archimedean ℓ-groups with distinguished weak order unit.

Monoid based semantics for linear formulas

2001 ◽  
Vol 66 (4) ◽  
pp. 1597-1619 ◽  
Author(s):  
W. P. R. Mitchell ◽  
H. Simmons
Keyword(s):  
Ordered Group ◽  
Partially Ordered ◽  
Ordered Monoid ◽  
Partially Ordered Group ◽  
Girard Quantale ◽  
Partially Ordered Monoid

Abstract.Each Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.

Monoid based semantics for linear formulas (corrected republication)

2002 ◽  
Vol 67 (2) ◽  
pp. 505-527
Author(s):  
W. P. R. Mitchell ◽  
H. Simmons
Keyword(s):  
Ordered Group ◽  
Partially Ordered ◽  
Ordered Monoid ◽  
Partially Ordered Group ◽  
Girard Quantale ◽  
Partially Ordered Monoid

AbstractEach Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.

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