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 X_{alpha+n }-free algebra of cardinality X_{alpha+n} from a suitable non-free X_{alpha}-free algebra, when is regular. The algebras constructed turn out to be elementarily equivalent in the language L_{infinityXalpha+n } to free algebras in the variety.
As applications of the technique, it is shown that for any positive integer n there are 2Xn X_{n}---free algebras which are generated X_{n} elements, cannot be generated by fewer than this number and are L_{infinityXn}-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 L_{infinitylambda} 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.
This is a Accepted version This version's date is: 1982 This item is not peer reviewed
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