On the number of finite algebraic structures

Aichinger, Erhard, Mayr, Peter, and McKenzie, R.. "On the number of finite algebraic structures." Journal of the European Mathematical Society 016.8 (2014): 1673-1686. <http://eudml.org/doc/277397>.

@article{Aichinger2014,
abstract = {We prove that every clone of operations on a finite set $A$, if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting $R$ for some finitary relation $R$ over $A$. It follows that for a fixed finite set $A$, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term.},
author = {Aichinger, Erhard, Mayr, Peter, McKenzie, R.},
journal = {Journal of the European Mathematical Society},
keywords = {Malcev conditions; few subpowers; term equivalence; clones; relations; Mal'tsev conditions; few subpowers; term equivalence; clones; relations},
language = {eng},
number = {8},
pages = {1673-1686},
publisher = {European Mathematical Society Publishing House},
title = {On the number of finite algebraic structures},
url = {http://eudml.org/doc/277397},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Aichinger, Erhard
AU - Mayr, Peter
AU - McKenzie, R.
TI - On the number of finite algebraic structures
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 8
SP - 1673
EP - 1686
AB - We prove that every clone of operations on a finite set $A$, if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting $R$ for some finitary relation $R$ over $A$. It follows that for a fixed finite set $A$, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term.
LA - eng
KW - Malcev conditions; few subpowers; term equivalence; clones; relations; Mal'tsev conditions; few subpowers; term equivalence; clones; relations
UR - http://eudml.org/doc/277397
ER -