# On the number of finite algebraic structures

Erhard Aichinger; Peter Mayr; R. McKenzie

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 8, page 1673-1686
- ISSN: 1435-9855

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topAichinger, 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 -

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