# Eilenberg-Moore algebra

From Encyclopedia of Mathematics

*Moore–Eilenberg algebra*

Given a monad (or triple) in a category , a -algebra is a pair , , , such that the diagram

commutes. Such a -algebra is also called an Eilenberg–Moore algebra. The forgetful functor from the category of Eilenberg–Moore algebras to has a left adjoint, exhibiting the monad as coming from a pair of adjoint functors (the Eilenberg–Moore construction).

See also Adjoint functor.

#### References

[a1] | F. Borceux, "Handbook of categorical algebra: Categories and structures" , 2 , Cambridge Univ. Press (1994) pp. Chap. 4 |

**How to Cite This Entry:**

Eilenberg-Moore algebra.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Eilenberg-Moore_algebra&oldid=17237

This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article