Page 269 - Textos de Matemática Vol. 46

ON THE LOCALNESS OF THE EMBEDDING OF ALGEBRAS
LURDES SOUSA
Dedicated to Manuela Sobral
Abstract. Let N : A ! B be a faithful functor between categories. Given an object B of B, we may ask whether there is an embedding B ! NA with A 2 A. In some cases the answer is well known. For instance, an abelian semigroup may be embedded in an abelian group if and only if it is cancellative. And every Lie algebra over a field K is embeddable in an associative K-algebra with identity. Many other examples are known. This paper concentrates on the localness of the embeddability. That is, it studies conditions under which the following property holds: B 2 B is embeddable in NA for some object A of A, whenever every finitely generated subobject of B is so.
1. Introduction
The following problem has been investigated for various algebraic categories: Let N : A ! B be a faithful functor; given an object B of B, determine if there is a monomorphism B ,! NA with A 2 A. The following two results on this subject are well known:
(a) An abelian semigroup may be embedded in an abelian group if and only if it is cancellative.
(b) Poincar´e-Birkho↵-Witt Theorem: Every Lie algebra over a field K is embeddable in an associative K-algebra with identity.
There are many other examples on the embeddability of algebras in the literature. J. MacDonald studied the subject from a categorical point of view
Received: 30 September 2014 / Accepted: 8 January 2015.
2010 Mathematics Subject Classification. 18B15, 03C05, 18C05.
Key words and phrases. Embedding theorems, categories of algebras, finitely generated
subobjects.
This work was partially supported by the Centro de Matema´tica da Universidade de
Coimbra (CMUC), funded by the European Regional Development Fund through the pro- gram COMPETE and by the Portuguese Government through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia under the projects PEst-C/MAT/UI0324/2013 and MCANA PTDC/MAT/120222/2010.
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