Abstract
In this paper the study of which varieties, in a countable similarity type, have non-free {Mathematical expression} (or equivalently א1-free) algebras is completed. It was previously known that if a variety satisfies a property known as the construction principle then there are such algebras. If a variety does not satisfy the construction principle then either every {Mathematical expression}-free algebra is free or for every infinite cardinal k, there is a k+-free algebra of cardinality k+ which is not free. Under the set theoretic assumption V=L, for any variety V in a countable similarity type, either the class of free algebras is definable in {Mathematical expression} or it is not definable in any {Mathematical expression}.
Original language | English |
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Pages (from-to) | 351-366 |
Number of pages | 16 |
Journal | Algebra Universalis |
Volume | 26 |
Issue number | 3 |
DOIs | |
State | Published - Oct 1989 |