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A [[Semi-group|semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n0667401.png" /> with zero for which there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n0667402.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n0667403.png" />; this is equivalent to the identity
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A [[Semi-group|semi-group]] $S$ with zero for which there is an $n$ such that $S^n = 0$; this is equivalent to the identity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n0667404.png" /></td> </tr></table>
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$$ x_1\dots x_n = y_1\dots y_n $$
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in $S$. The smallest $n$ with this property for a given semi-group is called the step (sometimes class) of nilpotency. If $S^2 = 0$, then $S$ is called a semi-group with zero multiplication. The following conditions on a semi-group $S$ are equivalent: 1) $S$ is nilpotent; 2) $S$ has a finite annihilator series (that is, an ascending annihilator series of finite length, see [[Nil semi-group|Nil semi-group]]); or 3) there is a $k$ such that every sub-semi-group of $S$ can be imbedded as an ideal series of length $\leq k$.
  
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n0667405.png" />. The smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n0667406.png" /> with this property for a given semi-group is called the step (sometimes class) of nilpotency. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n0667407.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n0667408.png" /> is called a semi-group with zero multiplication. The following conditions on a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n0667409.png" /> are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674010.png" /> is nilpotent; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674011.png" /> has a finite annihilator series (that is, an ascending annihilator series of finite length, see [[Nil semi-group|Nil semi-group]]); or 3) there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674012.png" /> such that every sub-semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674013.png" /> can be imbedded as an ideal series of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674014.png" />.
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A wider concept is that of a nilpotent semi-group in the sense of Mal'tsev [[#References|[2]]]. This is the name for a semi-group satisfying for some $n$ the identity
  
A wider concept is that of a nilpotent semi-group in the sense of Mal'tsev [[#References|[2]]]. This is the name for a semi-group satisfying for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674015.png" /> the identity
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$$ X_n = Y_n, $$
 
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where the words $X_n$ and $Y_n$ are defined inductively as follows: $X_0 = x$, $Y_0 = y$, $X_n = X_{n-1}u_nY_{n-1}$, $Y_n = Y_{n-1}u_nX_{n-1}$, where $x$, $y$ and $u_1,\dots ,u_n$ are variables. A group is a nilpotent semi-group in the sense of Mal'tsev if and only if it is nilpotent in the usual group-theoretical sense (see [[Nilpotent group|Nilpotent group]]), and the identity $X_n = Y_n$ is equivalent to the fact that its class of nilpotency is $\leq n$. Every cancellation semi-group satisfying the identity $X_n = Y_n$ can be imbedded in a group satisfying the same identity.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674016.png" /></td> </tr></table>
 
 
 
where the words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674018.png" /> are defined inductively as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674025.png" /> are variables. A group is a nilpotent semi-group in the sense of Mal'tsev if and only if it is nilpotent in the usual group-theoretical sense (see [[Nilpotent group|Nilpotent group]]), and the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674026.png" /> is equivalent to the fact that its class of nilpotency is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674027.png" />. Every cancellation semi-group satisfying the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066740/n06674028.png" /> can be imbedded in a group satisfying the same identity.
 
  
 
====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><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Nilpotent semi-groups"  ''Uchen. Zap. Ivanov. Gos. Ped. Inst.'' , '''4'''  (1953)  pp. 107–111  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.N. Shevrin,  "On the general theory of semi-groups"  ''Mat. Sb.'' , '''53''' :  3  (1961)  pp. 367–386  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.N. Shevrin,  "Semi-groups all sub-semi-groups of which are accessible"  ''Mat. Sb.'' , '''61''' :  2  (1963)  pp. 253–256  (In 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><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "Nilpotent semi-groups"  ''Uchen. Zap. Ivanov. Gos. Ped. Inst.'' , '''4'''  (1953)  pp. 107–111  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.N. Shevrin,  "On the general theory of semi-groups"  ''Mat. Sb.'' , '''53''' :  3  (1961)  pp. 367–386  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.N. Shevrin,  "Semi-groups all sub-semi-groups of which are accessible"  ''Mat. Sb.'' , '''61''' :  2  (1963)  pp. 253–256  (In Russian)</TD></TR></table>

Revision as of 10:44, 15 January 2012

A semi-group $S$ with zero for which there is an $n$ such that $S^n = 0$; this is equivalent to the identity

$$ x_1\dots x_n = y_1\dots y_n $$ in $S$. The smallest $n$ with this property for a given semi-group is called the step (sometimes class) of nilpotency. If $S^2 = 0$, then $S$ is called a semi-group with zero multiplication. The following conditions on a semi-group $S$ are equivalent: 1) $S$ is nilpotent; 2) $S$ has a finite annihilator series (that is, an ascending annihilator series of finite length, see Nil semi-group); or 3) there is a $k$ such that every sub-semi-group of $S$ can be imbedded as an ideal series of length $\leq k$.

A wider concept is that of a nilpotent semi-group in the sense of Mal'tsev [2]. This is the name for a semi-group satisfying for some $n$ the identity

$$ X_n = Y_n, $$ where the words $X_n$ and $Y_n$ are defined inductively as follows: $X_0 = x$, $Y_0 = y$, $X_n = X_{n-1}u_nY_{n-1}$, $Y_n = Y_{n-1}u_nX_{n-1}$, where $x$, $y$ and $u_1,\dots ,u_n$ are variables. A group is a nilpotent semi-group in the sense of Mal'tsev if and only if it is nilpotent in the usual group-theoretical sense (see Nilpotent group), and the identity $X_n = Y_n$ is equivalent to the fact that its class of nilpotency is $\leq n$. Every cancellation semi-group satisfying the identity $X_n = Y_n$ can be imbedded in a group satisfying the same identity.

References

[1] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[2] A.I. Mal'tsev, "Nilpotent semi-groups" Uchen. Zap. Ivanov. Gos. Ped. Inst. , 4 (1953) pp. 107–111 (In Russian)
[3] L.N. Shevrin, "On the general theory of semi-groups" Mat. Sb. , 53 : 3 (1961) pp. 367–386 (In Russian)
[4] L.N. Shevrin, "Semi-groups all sub-semi-groups of which are accessible" Mat. Sb. , 61 : 2 (1963) pp. 253–256 (In Russian)
How to Cite This Entry:
Nilpotent semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nilpotent_semi-group&oldid=13014
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article