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Difference between revisions of "Normal complex"

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''of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067420/n0674201.png" />''
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''of a semi-group $S$''
  
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)]]).
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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)|Congruence (in algebra)]]).
  
 
====References====
 
====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>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.S. Lyapin,  "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR></table>

Latest revision as of 20:27, 14 April 2014

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