# Normal complex

From Encyclopedia of Mathematics

*of a semi-group $S$*

A non-empty subset $N\subseteq S$ satisfying the following condition: For any $x,y\in S^1$ (where $S^1=S$ when $S$ contains a unit element and $S^1$ is the semi-group obtained from $S$ by adjoining a unit element if $S$ does not have one) and any $a,b\in N$ it follows from $xay\in N$ that $xby\in N$. A subset $N$ is a normal complex of a semi-group $S$ if and only if $N$ is a class of some congruence on $S$ (cf. Congruence (in algebra)).

#### References

[1] | E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) |

**How to Cite This Entry:**

Normal complex.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Normal_complex&oldid=31680

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