Variety

A variety is a class of algebras that is closed under homomorphisms, subalgebras, and direct products. Examples include the variety of groups, the variety of rings, the variety of lattices. The class of fields (viewed as a subclass of the class of rings) is not a variety, because it is not closed under direct products.

Some important varieties, such as the variety of distributive lattices, are locally finite, meaning that their finitely generated algebras are finite. Others, such as the variety of all lattices, are not locally finite. In strong varieties, direct sums of locally finite algebras are locally finite.

Note that this type of variety arises in universal algebra and really has nothing to do with algebraic varieties, toric varieties, etc.

REFERENCES

Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981.

 http://www.thoralf.uwaterloo.ca/htdocs/ualg.html.Cohn, P. M. Universal Algebra. New York: Harper and Row, 1965.

Grätzer, G. Universal Algebra, 2nd ed. New York: Springer-Verlag, 1979.

Insall, M. "Nonstandard Methods and Finiteness Conditions in Algebra." Zeitschrifte für Math. Logik und Grundlagen d. Math. 37, 525-532, 1991.