# Unitization of Ball truncated $${\ell}$$ ℓ -groups

Unitization of Ball truncated $${\ell}$$ ℓ -groups Recently, Ball defined a truncated $${\ell}$$ ℓ -group to be an $${\ell}$$ ℓ -group G along with a truncation. We constructively prove that if G is a truncated $${\ell}$$ ℓ -group, then the direct sum $${G \oplus \mathbb{Q}}$$ G ⊕ Q is equipped with a structure of an $${\ell}$$ ℓ -group with weak unit the rational number 1. As a simple consequence, we get a description of the truncated $${\ell}$$ ℓ - group obtained by Ball via representation theory. On the other hand, we derive some characterizations of truncation morphisms as defined by Ball himself. In particular, we show that the group homomorphism $${f : G \rightarrow H}$$ f : G → H is a truncation morphism if and only its natural extension $${f^*}$$ f ∗ from $${G \oplus \mathbb{Q}}$$ G ⊕ Q into $${H \oplus \mathbb{Q}}$$ H ⊕ Q is an $${\ell}$$ ℓ -homomorphism. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png algebra universalis Springer Journals

# Unitization of Ball truncated $${\ell}$$ ℓ -groups

, Volume 78 (1) – May 22, 2017
12 pages

/lp/springer_journal/unitization-of-ball-truncated-ell-groups-bKmb9GnWii
Publisher
Springer International Publishing
Subject
Mathematics; Algebra
ISSN
0002-5240
eISSN
1420-8911
D.O.I.
10.1007/s00012-017-0444-1
Publisher site
See Article on Publisher Site

### Abstract

Recently, Ball defined a truncated $${\ell}$$ ℓ -group to be an $${\ell}$$ ℓ -group G along with a truncation. We constructively prove that if G is a truncated $${\ell}$$ ℓ -group, then the direct sum $${G \oplus \mathbb{Q}}$$ G ⊕ Q is equipped with a structure of an $${\ell}$$ ℓ -group with weak unit the rational number 1. As a simple consequence, we get a description of the truncated $${\ell}$$ ℓ - group obtained by Ball via representation theory. On the other hand, we derive some characterizations of truncation morphisms as defined by Ball himself. In particular, we show that the group homomorphism $${f : G \rightarrow H}$$ f : G → H is a truncation morphism if and only its natural extension $${f^*}$$ f ∗ from $${G \oplus \mathbb{Q}}$$ G ⊕ Q into $${H \oplus \mathbb{Q}}$$ H ⊕ Q is an $${\ell}$$ ℓ -homomorphism.

### Journal

algebra universalisSpringer Journals

Published: May 22, 2017

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