scholarly journals The Hahn Embedding Theorem for a Class of Residuated Semigroups

Studia Logica ◽  
2020 ◽  
Vol 108 (6) ◽  
pp. 1161-1206
Author(s):  
Sándor Jenei
Keyword(s):  
Representation Theorem ◽  
Abelian Groups ◽  
Embedding Theorem ◽  
Residuated Lattices ◽  
Lexicographic Product ◽  
Product Construction

AbstractHahn’s embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn’s theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial lexicographic product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains which possess only finitely many idempotent elements, by means of linearly ordered abelian groups and the partial lexicographic product construction is presented.

A generalization of co-Hopfian abelian groups

2017 ◽  
Vol 27 (04) ◽  
pp. 351-360
Author(s):  
Andrey Chekhlov ◽  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Representation Theorem ◽  
Abelian Groups ◽  
Classical Notion ◽  
Group A ◽  
Hopfian Group ◽  
Comprehensive Study

We define the concept of an [Formula: see text]-co-Hopfian abelian group, which is a nontrivial generalization of the classical notion of a co-Hopfian group. A systematic and comprehensive study of these groups is given in very different ways. Specifically, a representation theorem for [Formula: see text]-co-Hopfian groups is established as well as it is shown that there exists an [Formula: see text]-co-Hopfian group which is not co-Hopfian.

scholarly journals A representation theorem for integral rigs and its applications to residuated lattices

2016 ◽  
Vol 220 (10) ◽  
pp. 3533-3566 ◽  
Author(s):  
J.L. Castiglioni ◽  
M. Menni ◽  
W.J. Zuluaga Botero
Keyword(s):  
Representation Theorem ◽  
Residuated Lattices

Valued Abelian Groups

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Abelian Groups ◽  
Logarithmic Derivative ◽  
General Setting ◽  
Embedding Theorem ◽  
Vector Spaces ◽  
Ordered Sets ◽  
Valued Fields ◽  
Differential Field ◽  
Basic Facts

This chapter deals with valued abelian groups. It first introduces some terminology concerning ordered sets before discussing valued abelian groups and ordered abelian groups in more detail. Ordered abelian groups occur as value groups of valued fields, whereas valued abelian groups arise because the logarithmic derivative map on a valued differential field like induces a valuation on the value group that turns out to be very useful. Furthermore, the notion of a pseudocauchy sequence makes perfect sense in the general setting of valued abelian groups, and the basic facts about these sequences yield a natural proof of a generalized Hahn Embedding Theorem. The chapter also considers valued vector spaces, including spherically complete valued vector spaces, and proves a version of the Hahn Embedding Theorem for valued vector spaces. Special attention is given to particularly well-behaved valued vector spaces known as Hahn spaces.

On Teh's construction of orders in Abelian groups

1972 ◽  
Vol 71 (3) ◽  
pp. 433-436
Author(s):  
Donald P. Minassian
Keyword(s):  
Direct Product ◽  
Abelian Groups ◽  
Rational Numbers ◽  
Embedding Theorem

AbstractA claim of H. H. Teh on Archimedean full orders ((2), ‘only if’ of Theorem 2, p. 480) fails. His statement (p. 478, line 5) that the direct product of n copies of the additive rational numbers admits no full order of Archimedean rank more than n is proved without Hahn's embedding theorem. A few other remarks are made.

scholarly journals Inner Product Groups and Riesz Representation Theorem

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1946
Author(s):  
Alireza Pourmoslemi ◽  
Tahereh Nazari ◽  
Mehdi Salimi
Keyword(s):  
Representation Theorem ◽  
Abelian Groups ◽  
Inner Product ◽  
Riesz Representation Theorem ◽  
Product Group ◽  
Basic Properties ◽  
Inner Products ◽  
Normed Group ◽  
Riesz Representation ◽  
Torsion Free

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

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