View source for Normal complex

''of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674201.png" />''

A non-empty subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674202.png" /> satisfying the following condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674203.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674204.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674205.png" /> contains a unit element and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674206.png" /> is the semi-group obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674207.png" /> by adjoining a unit element if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674208.png" /> does not have one) and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674209.png" /> it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n06742010.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n06742011.png" />. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n06742012.png" /> is a normal complex of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n06742013.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n06742014.png" /> is a class of some congruence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n06742015.png" /> (cf. [[Congruence (in algebra)|Congruence (in algebra)]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR></table>