PreprintCC BY-NC-SA 4.0arXiv
Properties of theories whose algebras are free
Abstract
A hard theorem of Givant classifies the finitary algebraic theories for which every algebra is free. From it, one easily deduces that all such theories are minimal (adding any further equations makes them trivial) and have a well-defined notion of dimension (free algebras on sets of different cardinality are never isomorphic). We prove both these properties directly, without the aid of Givant's theorem.
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BibTeX
@misc{leinster2026properties,
title = {{Properties of theories whose algebras are free}},
author = {Tom Leinster and Maia Woolf},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31599},
eprint = {2610.31599},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31599},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Properties of theories whose algebras are free
AU - Tom Leinster
AU - Maia Woolf
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31599
UR - https://arxiv.org/abs/2610.31599
AB - A hard theorem of Givant classifies the finitary algebraic theories for which every algebra is free. From it, one easily deduces that all such theories are minimal (adding any further equations makes them trivial) and have a well-defined notion of dimension (free algebras on sets of different cardinality are never isomorphic). We prove both these properties directly, without the aid of Givant's theorem.
ER -