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Difference between revisions of "Normal sub-semi-group"

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''of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n0676801.png" />''
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''of a semi-group $S$''
  
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A sub-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n0676802.png" /> satisfying the following condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n0676803.png" /> (for the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n0676804.png" /> see [[Normal complex|Normal complex]]) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n0676805.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n0676806.png" /> the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n0676807.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n0676808.png" /> are equivalent. A subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n0676809.png" /> is a normal sub-semi-group if and only if it is the complete inverse image of the unit element under some homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067680/n06768010.png" /> onto a [[Semi-group|semi-group]] with unit element.
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A sub-semi-group $H$ satisfying the following condition: For any $x,y\in S^1$ (for the notation $S^1$ see [[Normal complex|Normal complex]]) such that $xy\in S$ and for any $h\in H$ the relations $xhy\in H$ and $xy\in H$ are equivalent. A subset of $S$ is a normal sub-semi-group if and only if it is the complete inverse image of the unit element under some homomorphism of $S$ onto a [[Semi-group|semi-group]] with unit element.
  
 
====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>

Revision as of 14:14, 1 May 2014

of a semi-group $S$

A sub-semi-group $H$ satisfying the following condition: For any $x,y\in S^1$ (for the notation $S^1$ see Normal complex) such that $xy\in S$ and for any $h\in H$ the relations $xhy\in H$ and $xy\in H$ are equivalent. A subset of $S$ is a normal sub-semi-group if and only if it is the complete inverse image of the unit element under some homomorphism of $S$ onto a semi-group with unit element.

References

[1] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
How to Cite This Entry:
Normal sub-semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_sub-semi-group&oldid=32024
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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