Infinitary properties of valued and ordered vector spaces

In this chapter, we discuss several special types of modules that will make frequent appearances later. These include semisimple modules, artinian modules, and torsionfree modules, which have prototypes in vector spaces, finite abelian groups, and torsionfree abelian groups, respectively. That special types of modules have useful roles in the study of arbitrary modules may be seen already in the case of abelian groups (i.e., ℤ-modules). In studying an arbitrary abelian group, an almost reflexive first step is to look at its torsion part (i.e., the torsion subgroup) and its torsionfree part (i.e., the factor group modulo the torsion subgroup), since entirely different techniques are available (and needed) for dealing with torsion groups and torsionfree groups. On the torsionfree side, vector spaces make an appearance due to the facts that the torsionfree divisible abelian groups are exactly the vector spaces over ℚ and that every torsionfree abelian group can be embedded in a divisible one. On the torsion side, many questions can be reduced to the case of finite abelian groups, since every torsion abelian group is a directed union of finite subgroups. In studying torsion abelian groups, one also reduces to the case of p -groups for various primes p . Vector spaces make another appearance here, since in an abelian p -group the set of elements of order p (together with 0) forms a vector space over ℤ/ p ℤ.

Bounded subgroups as a von Neumann radical of an Abelian group

Fuzzy topological ordered vector spaces I

Minimal sumsets in finite solvable groups

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, provingTheorem. Every group of bounded order is a direct sum of cyclic groups, andTheorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.

In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments \( \{ z \in V:a < z < b\} \), a < b, is a base for the Euclidean topology).

We present a general algebraic basis for arbitrary systems of units such as those used in physical sciences, engineering, and economics. Physical quantities are represented as q-numbers: an ordered pair u = {u,labelu}, that is, u ∈ q = X × W B . The algebraic structure of the infinite sets of labels that represent the “units” has been established: such sets W B are infinite Abelian multiplicative groups with a finite basis. W B is solvable as it admits a tower of Abelian subgroups. Extensions to include the possibility of rational powers of labels have been included, as well as the addition of named labels. Named labels are an essential feature of all practical systems of units. Furthermore, q is an Abelian multiplicative group, and it is not a ring. q admits decomposition into one-dimensional normed vector spaces over the field X among members with equivalent labels. These properties lead naturally to the concept of well-posed relations, and to Buckingham’s theorem of dimensional analysis. Finally, a connection is made with a Group Ring structure and an interpretation in terms of the observable properties of physico-chemical systems is given.

If the vector space \({\mathcal{L}}^r(E, F)\) of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for \({\mathcal{L}}^r(E, F)\) to be a vector lattice. \({\mathcal{L}}^r(E, F)\) and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces \({\mathcal{V}}(E, F)\) of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space \({\mathcal{L}}^r(E, F)\) or, in the case that \({\mathcal{L}}^r(E, F)\) is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices \({\mathcal{V}}(E, F)\), where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and \({\mathcal{V}}(E, F)\) are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in \({\mathcal{V}}(E,F)\). The paper contains also some generalization of results obtained for the case \({\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)\) in [10].

Abstract.A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably-generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tuples depend only on the Ulm invariants of the module in question (and not on the underlying ring). A complete description of the poset of orbits of elements in terms of the Ulm invariants of the module is given. The relationship between this description of orbits and a very different-looking one obtained by Dutta and Prasad for torsion modules of bounded order is explained.

If E is a ring, and G is an E-module, then G is quasi-injective if every homomorphism from a submodule of G to G can be extended to an endomorphism of G. POOLE and REID [2] raise the question as to which abelian groups G are quasiinjective as modules over their endomorphism rings E. They prove that direct sums of cyclic p-groups are quasi-injective. This result is extended here to a large class of p-groups, including all p-groups that have no elements of infinite height, and all totally projective p-groups -- in particular, all countable p-groups. The central idea is a property shared by totally projective p-groups and pgroups with no elements of infinite height. DEFINITION. If G is a p-group and xEG[p], then x is said to be detachable if G can be written as G1 @ Gz with (x) =p'G~ [p] for some ordinal c~. If every element of G[p] is detachable, we say that G is detachable. We write p'G~[p] instead of p~G~ to allow Ga to be divisible. Note that every element in G[p] of finite height is detachable, so every p-group with no elements of infinite height is detachable. Hill's extension of Ulm's theorem to totally projective p-groups (see [3]) implies that these groups are also detachable. THEOREM 1. If G is a detachable p-group, and x, yEG[p], then there is an endomorphism f of G such that f(x) =y or f(y) = x.

This chapter is devoted to studying some concepts that will be extensively used in the last chapters, namely the cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions (which are related to second cohomology groups). The same three different cases of extensions of chapter 5 as well as the ℱ( M )-valued version of cohomology will be considered. In fact, the relation between Lie group and Lie algebra cohomology will be explored here, first with the simple example of central extensions of groups and algebras (governed by twococycles), and then in the higher order case, providing explicit formulae for obtaining Lie algebra cocycles from Lie group ones and vice versa. The Lie algebra cohomology a la Chevalley-Eilenberg, which uses invariant forms on a Lie group, is also presented in this chapter. This will turn out to be specially useful in the construction of physical actions (chapter 8), i.e . in the process of relating cohomology and mechanics. The BRST formulation of Lie algebra cohomology will also be discussed here due to its importance in gauge theories. Cohomology of Lie algebras: general definitions We now discuss the cohomology of Lie algebras with values on a vector space V following the same pattern as used in chapter 5 for the cohomology of groups with values in an abelian group. Let G be a Lie algebra, and let the vector space V over the field K be a ρ(G)-module.

In this paper, we present a method to create a safe arithmetic that can be used to obfuscate implementations that require operations over commutative groups. The method is based on the structure of the endomorphisms of certain extensions of the original commutative group. The endomorphisms of a commutative group are non-commutative (in general), thus we can use a non-commutative group to emulate the arithmetic of a commutative one. The techniques presented in this paper are very flexible and the programmer has a wide variety of options to obfuscate the algorithms. The system can be parameterized using conjugations, thus it is possible to generate a different arithmetic for each instance of the program with a change in the security parameters, even in cases in which this number is huge (for example, in IoT applications). The security of this method is based not only on the difficulty of the conjugacy search problem (in a harder version because only partial information about the groups is known by the attacker), but also in a number of extra options that can be chosen by the programmer. The paper explains the general method, analyzes its algebraic properties and provides detailed examples based on the vector spaces over and XOR operators.

This paper demonstrates how a Political Economy of Wealth – an analytical framework inspired from Ricardo's and Marx's theories of value – strengthens the analytical force of Socio-Ecological Economics in the context of the controversy over the value of nature. The Political Economy of Wealth helps (1) to overcome some theoretical limitations encountered in Socio-Ecological Economics, (2) to develop a critical perspective on neoclassical theory of environmental values, as well as a new justification of value incommensurability, and (3) to move toward a new research agenda that aims to study interactions between the economy and the environment from a socio-historical perspective.

Vector spaces over unspecified fields can be axiomatized as one-sorted structures, namely, abelian groups with the relation of parallelism. Parallelism is binary linear dependence. When equipped with the n-ary relation of linear dependence for some positive integer n, a vector-space is existentially closed if and only if it is n-dimensional over an algebraically closed field. In the signature with an n-ary predicate for linear dependence for each positive integer n, the theory of infinite-dimensional vector spaces over algebraically closed fields is the model-completion of the theory of vector spaces.

We consider the class of abelian groups possessing partial decomposition bases in L ∞ ω δ L^\delta _{\infty \omega } for δ \delta an ordinal. This class contains the class of Warfield groups which are direct summands of simply presented groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. We prove a classification theorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Göbel and the authors.