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''of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i0500601.png" />''
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{{TEX|done}}
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''of a semi-group $S$''
  
 
A sequence of sub-semi-groups
 
A sequence of sub-semi-groups
  
<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/i/i050/i050060/i0500602.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$A_1\subset A_2\subset\ldots\subset A_m=S\tag{*}$$
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i0500603.png" /> is a (two-sided) ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i0500604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i0500605.png" />. The sub-semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i0500606.png" /> and the Rees factor semi-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i0500607.png" /> (see [[Semi-group|Semi-group]]) are called the factors of the series (*). Two ideal series are said to be isomorphic if a one-to-one correspondence can be established between the factors such that corresponding factors are isomorphic. An ideal series
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such that $A_i$ is a (two-sided) ideal of $A_{i+1}$, $i=1,\dots,m-1$. The sub-semi-group $A_1$ and the Rees factor semi-groups $A_{i+1}/A_i$ (see [[Semi-group|Semi-group]]) are called the factors of the series \ref{*}. Two ideal series are said to be isomorphic if a one-to-one correspondence can be established between the factors such that corresponding factors are isomorphic. An ideal series
  
<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/i/i050/i050060/i0500608.png" /></td> </tr></table>
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$$B_1\subset B_2\subset\ldots\subset B_n=S$$
  
is said to be a refinement of (*) if every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i0500609.png" /> occurs among the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i05006010.png" />. An ideal series is a composition series if it does not have proper refinements. Any two ideal series of a semi-group have isomorphic refinements; in particular, in a semi-group having a composition series all such series are isomorphic (the analogue of the theorems of Schreier and Jordan–Hölder for normal series in groups, see [[#References|[1]]], [[#References|[2]]]). An ideal series is a chief series if its terms are ideals in the whole semi-group and if it has no proper refinements consisting of ideals of the semi-group. If a semi-group has a composition series, then it also has a chief series; the converse is false. In a semi-group with a chief series, its factors are isomorphic to the chief factors (cf. [[Principal factor|Principal factor]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i05006011.png" />.
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is said to be a refinement of \ref{*} if every $A_i$ occurs among the $B_j$. An ideal series is a composition series if it does not have proper refinements. Any two ideal series of a semi-group have isomorphic refinements; in particular, in a semi-group having a composition series all such series are isomorphic (the analogue of the theorems of Schreier and Jordan–Hölder for normal series in groups, see [[#References|[1]]], [[#References|[2]]]). An ideal series is a chief series if its terms are ideals in the whole semi-group and if it has no proper refinements consisting of ideals of the semi-group. If a semi-group has a composition series, then it also has a chief series; the converse is false. In a semi-group with a chief series, its factors are isomorphic to the chief factors (cf. [[Principal factor|Principal factor]]) of $S$.
  
As for normal series in groups, the concepts mentioned above (as well as their properties) naturally generalize to the case of infinite systems of nested sub-semi-groups. In particular, an ascending ideal series in a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i05006012.png" /> is a totally ordered sequence
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As for normal series in groups, the concepts mentioned above (as well as their properties) naturally generalize to the case of infinite systems of nested sub-semi-groups. In particular, an ascending ideal series in a semi-group $S$ is a totally ordered sequence
  
<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/i/i050/i050060/i05006013.png" /></td> </tr></table>
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$$A_1\subset\ldots\subset A_\alpha\subset A_{\alpha+1}\subset\ldots\subset A_\beta=S,$$
  
where at limit points there stand the unions of the preceding members, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i05006014.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i05006015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050060/i05006016.png" />.
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where at limit points there stand the unions of the preceding members, and $A_\alpha$ is an ideal of $A_{\alpha+1}$ for all $\alpha<\beta$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR></table>

Revision as of 11:43, 9 November 2014

of a semi-group $S$

A sequence of sub-semi-groups

$$A_1\subset A_2\subset\ldots\subset A_m=S\tag{*}$$

such that $A_i$ is a (two-sided) ideal of $A_{i+1}$, $i=1,\dots,m-1$. The sub-semi-group $A_1$ and the Rees factor semi-groups $A_{i+1}/A_i$ (see Semi-group) are called the factors of the series \ref{*}. Two ideal series are said to be isomorphic if a one-to-one correspondence can be established between the factors such that corresponding factors are isomorphic. An ideal series

$$B_1\subset B_2\subset\ldots\subset B_n=S$$

is said to be a refinement of \ref{*} if every $A_i$ occurs among the $B_j$. An ideal series is a composition series if it does not have proper refinements. Any two ideal series of a semi-group have isomorphic refinements; in particular, in a semi-group having a composition series all such series are isomorphic (the analogue of the theorems of Schreier and Jordan–Hölder for normal series in groups, see [1], [2]). An ideal series is a chief series if its terms are ideals in the whole semi-group and if it has no proper refinements consisting of ideals of the semi-group. If a semi-group has a composition series, then it also has a chief series; the converse is false. In a semi-group with a chief series, its factors are isomorphic to the chief factors (cf. Principal factor) of $S$.

As for normal series in groups, the concepts mentioned above (as well as their properties) naturally generalize to the case of infinite systems of nested sub-semi-groups. In particular, an ascending ideal series in a semi-group $S$ is a totally ordered sequence

$$A_1\subset\ldots\subset A_\alpha\subset A_{\alpha+1}\subset\ldots\subset A_\beta=S,$$

where at limit points there stand the unions of the preceding members, and $A_\alpha$ is an ideal of $A_{\alpha+1}$ for all $\alpha<\beta$.

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
How to Cite This Entry:
Ideal series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ideal_series&oldid=17581
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article