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Difference between revisions of "Gauss semi-group"

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A commutative semi-group with unit satisfying the cancellation law, in which any non-invertible element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043530/g0435301.png" /> is decomposable into a product of irreducible elements (i.e. elements that cannot be represented as a product of non-invertible factors); moreover, for each two such decompositions
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A commutative semi-group with unit satisfying the cancellation law, in which any non-invertible element $a$ is decomposable into a product of irreducible elements (i.e. elements that cannot be represented as a product of non-invertible factors); moreover, for each two such decompositions
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$$
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a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l
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$$
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one has $k=l$ and, possibly after renumbering the factors, also
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$$
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b_1 = c_1 \epsilon_1,\ \ldots,\ b_k = c_k \epsilon_k
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$$
  
<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/g/g043/g043530/g0435302.png" /></td> </tr></table>
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where $\epsilon_1,\ldots,\epsilon_k$ are invertible elements. Typical examples of Gauss semi-groups include the multiplicative semi-group of non-zero integers, and that of non-zero polynomials in one unknown over a field. Any two elements of a Gauss semi-group have a highest common divisor.
 
 
one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043530/g0435303.png" /> and, possibly after renumbering the factors, also
 
 
 
<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/g/g043/g043530/g0435304.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043530/g0435305.png" /> are invertible elements. Typical examples of Gauss semi-groups include the multiplicative semi-group of non-zero integers, and that of non-zero polynomials in one unknown over a field. Any two elements of a Gauss semi-group have a highest common divisor.
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>

Revision as of 18:56, 12 October 2014

A commutative semi-group with unit satisfying the cancellation law, in which any non-invertible element $a$ is decomposable into a product of irreducible elements (i.e. elements that cannot be represented as a product of non-invertible factors); moreover, for each two such decompositions $$ a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l $$ one has $k=l$ and, possibly after renumbering the factors, also $$ b_1 = c_1 \epsilon_1,\ \ldots,\ b_k = c_k \epsilon_k $$

where $\epsilon_1,\ldots,\epsilon_k$ are invertible elements. Typical examples of Gauss semi-groups include the multiplicative semi-group of non-zero integers, and that of non-zero polynomials in one unknown over a field. Any two elements of a Gauss semi-group have a highest common divisor.

References

[1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
How to Cite This Entry:
Gauss semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_semi-group&oldid=33585
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article