Some applications of set theory to algebra

Pope, Alun Lloyd


Pope, Alun Lloyd (1982) Some applications of set theory to algebra.

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This thesis deals with two topics.

In Part I it is shown that if ZFC is consistent, then so is ZF + the order extension principle + there is an abelian group without a divisible hull. The proof is by forcing.

In Part II a technique is developed which, in many varieties of algebras, enables the construction for each positive integer not a non-free Xalpha+n -free algebra of cardinality Xalpha+n from a suitable non-free Xalpha-free algebra, when is regular. The algebras constructed turn out to be elementarily equivalent in the language LinfinityXalpha+n to free algebras in the variety.

As applications of the technique, it is shown that for any positive integer n there are 2Xn Xn---free algebras which are generated Xn elements, cannot be generated by fewer than this number and are LinfinityXn-equvalent to free algebras in each of the following varieties: any torsion-free variety of groups, all rings with a 1, all commutative rings with a 1, all K-algebras (with K a not-necessarily commutative integral domain), all Lie algebras over a given field.

By a different analysis it is shown too that in any variety of nilpotent groups, a lambda-free group of uncountable cardinality lambda is free (respectively, equivalent in Linfinitylambda to a free group) if and only if its abelianisation is, in the abelian part of the variety.

Finally, sufficient conditions are given for a X-free group in a variety of groups to be also para free in the variety. The results imply that in the varieties of all groups soluble of length at most k and of all groups polynil potent of given class, if lambda is singular or weakly compact, then a lambda-free group of cardinality lambda is parafree, while if lambda is strongly compact, then a lambda-free group of any cardinality is parafree.

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This is a Accepted version
This version's date is: 1982
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Item TypeThesis (Doctoral)
TitleSome applications of set theory to algebra
AuthorsPope, Alun Lloyd
Uncontrolled KeywordsMathematics; Pure Sciences; Algebra; Applications; Algebra; Set; Set Theory; Some; Set Theory; Theory



Deposited by () on 01-Feb-2017 in Royal Holloway Research Online.Last modified on 01-Feb-2017


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