scholarly journals Kernels in tropical geometry and a Jordan–Hölder theorem

2018 ◽  
Vol 17 (04) ◽  
pp. 1850066 ◽  
Author(s):  
Tal Perri ◽  
Louis H. Rowen
Keyword(s):  
Tropical Geometry ◽  
Rational Functions ◽  
Composition Series ◽  
Dimension Theory ◽  
Ordered Group ◽  
Additive Structure ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Roots Of Polynomials ◽  
Tropical Varieties

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, [Formula: see text]-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield[Formula: see text][Formula: see text], we pass to the semifield[Formula: see text][Formula: see text] of fractions of the polynomial semiring[Formula: see text], for which there already exists a well developed theory of kernels, which are normal convex subgroups of [Formula: see text]; the parallel of the zero set now is the [Formula: see text]-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to [Formula: see text]-kernels (Definition 4.1.4) and [Formula: see text]-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The [Formula: see text]-kernels corresponding to tropical hypersurfaces are the [Formula: see text]-sets of what we call “corner internal rational functions,” and we describe [Formula: see text]-kernels corresponding to “usual” tropical geometry as [Formula: see text]-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of [Formula: see text]-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between [Formula: see text]-sets and a class of [Formula: see text]-kernels of the rational [Formula: see text]-semifield[Formula: see text] called polars, originating from the theory of lattice-ordered groups. When [Formula: see text] is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal [Formula: see text]-kernels, intersected with the [Formula: see text]-kernel generated by [Formula: see text]. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of [Formula: see text]-kernels.

scholarly journals Minimal vector lattice covers

1971 ◽  
Vol 5 (3) ◽  
pp. 331-335 ◽  
Author(s):  
Roger D. Bleier
Keyword(s):  
Vector Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

Archimedean Closures in Lattice-Ordered Groups

1969 ◽  
Vol 21 ◽  
pp. 1004-1012 ◽  
Author(s):  
Richard D. Byrd
Keyword(s):  
Partial Answer ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Regular Subgroup

Conrad (10) and Wolfenstein (15; 16) have introduced the notion of an archimedean extension (a-extension) of a lattice-ordered group (l-group). In this note the class of l-groups that possess a plenary subset of regular subgroups which are normal in the convex l-subgroups that cover them are studied. It is shown in § 3 (Corollary 3.4) that the class is closed with respect to a-extensions and (Corollary 3.7) that each member of the class has an a-closure. This extends (6, p. 324, Corollary II; 10, Theorems 3.2 and 4.2; 15, Theorem 1) and gives a partial answer to (10, p. 159, Question 1). The key to proving both of these results is Theorem 3.3, which asserts that if a regular subgroup is normal in the convex l-subgroup that covers it, then this property is preserved by a-extensions.

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°).

On a relative uniform completion of an archimedean lattice ordered group

2009 ◽  
Vol 59 (2) ◽  
Author(s):  
Štefan Černák ◽  
Judita Lihová
Keyword(s):  
Uniform Convergence ◽  
Subdirect Product ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice ◽  
Finite Direct Product ◽  
Relatively Uniform Convergence

AbstractThe 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) $$ G_{\omega _1 } $$ of G is dealt with. It is known that $$ G_{\omega _1 } $$ 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 $$ G_{\omega _1 } $$. Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence.

Relatively uniform convergences in archimedean lattice ordered groups

2010 ◽  
Vol 60 (4) ◽  
Author(s):  
Ján Jakubík ◽  
Štefan Černák
Keyword(s):  
Uniform Convergence ◽  
Vector Lattice ◽  
Radical Class ◽  
Ordered Group ◽  
Dedekind Completion ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Divisible Hull ◽  
Archimedean Lattice ◽  
Relatively Uniform Convergence

AbstractFor 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.

scholarly journals C*-algebras generated by isometries and true representations

2019 ◽  
Vol 38 (5) ◽  
pp. 215-232
Author(s):  
Mamoon Ahmed
Keyword(s):  
The Other ◽  
Lattice Ordered Group ◽  
Covariant Representation ◽  
Ordered Group ◽  
Two Stage ◽  
C Algebras ◽  
Ordered Groups ◽  
Lattice Ordered Groups

Let (G; P) be a quasi-lattice ordered group. In this paper we present a modied proof of Laca and Raeburn's theorem about the covariant isometric representations of amenable quasi-lattice ordered groups [7, Theorem 3.7], by following a two stage strategy. First, we construct a universal covariant representation for a given quasi-lattice ordered group (G; P) and show that it is unique. The construction of this object is new; we have not followed either Nica's approach in [10] or Laca and Raeburn's approach in [7], although all three objects are essentially the same. Our approach is a very natural one and avoids some of the intricacies of the other approaches. Then we show if (G; P) is amenable, true representations of (G; P) generate C-algebras which are canonically isomorphic to the universal object.

Weak relatively uniform convergences on abelian lattice ordered groups

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
Štefan Černák ◽  
Ján Jakubík
Keyword(s):  
Uniform Convergence ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Abelian Lattice Ordered Group ◽  
Relative Uniform Convergence ◽  
Brouwerian Lattice ◽  
Abelian Lattice ◽  
Relatively Uniform Convergence

AbstractThe 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.

On a type of distributivity of lattice ordered groups

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Distributive Lattice ◽  
Boolean Algebras ◽  
Radical Class ◽  
Infinite Cardinal ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Homogeneity Condition ◽  
Lattice Ordered Groups

AbstractLet m be an infinite cardinal. Inspired by a result of Sikorski on m-representability of Boolean algebras, we introduce the notion of r m-distributive lattice ordered group. We prove that the collection of all such lattice ordered groups is a radical class. Using the mentioned notion, we define and investigate a homogeneity condition for lattice ordered groups.

scholarly journals l-simple lattice-ordered groups

1974 ◽  
Vol 19 (2) ◽  
pp. 133-138 ◽  
Author(s):  
A. M. W. Glass
Keyword(s):  
Group Theory ◽  
Lattice Ordered Group ◽  
Normal Subgroups ◽  
Ordered Group ◽  
Central Task ◽  
Ordered Groups ◽  
Simple Lattice ◽  
Lattice Ordered Groups ◽  
Group Play

Let G be a lattice-ordered group (l-group) and H a subgroup of G. H is said to be an l-subgroup of G if it is a sublattice of G. H is said to be convex if h1, h2 ∈ H and h2 ≦ g ≦ h2 imply g ∈ H. The normal convex l-subgroups (l-ideals) of an l-group play the same role in the study of lattice-ordered groups as do normal subgroups in the investigation of groups. For this reason, an l-group is said to be l-simple if it has no non-trivial l-ideals. As in group theory, a central task in the examination of lattice-ordered groups is to characterise those l-groups which are l-simple.

The Hahn representation theorem for ℓ-groups in ZFA

2000 ◽  
Vol 65 (2) ◽  
pp. 519-524
Author(s):  
D. Gluschankof
Keyword(s):  
Set Theory ◽  
Representation Theorem ◽  
Group Structure ◽  
Ordered Group ◽  
Representation Theorems ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Totally Ordered Group ◽  
Basic Concepts ◽  
The Axiom Of Choice

In [7] the author discussed the relative force —in the set theory ZF— of some representation theorems for ℓ-groups (lattice-ordered groups). One of the theorems not discussed in that paper is the Hahn representation theorem for abelian ℓ-groups. This result, originally proved by Hahn (see [8]) for totally ordered groups and half a century later by Conrad, Harvey and Holland for the general case (see [4]), states that any abelian ℓ-group can be embedded in a Hahn product of copies of R (the real line with its natural totally-ordered group structure). Both proofs rely heavily on Zorn's Lemma which is equivalent to AC (the axiom of choice).The aim of this work is to point out the use of non-constructible axioms (i.e., AC and weaker forms of it) in the proofs. Working in the frame of ZFA, that is, the Zermelo-Fraenkel set theory where a non-empty set of atoms is allowed, we present alternative proofs which, in the totally ordered case, do not require the use of AC. For basic concepts and notation on ℓ-groups the reader can refer to [1] and [2]. For set theory, to [11].

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