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Difference between revisions of "O-direct union"

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''of semi-groups with zero''
 
''of semi-groups with zero''
  
The [[Semi-group|semi-group]] obtained from the given family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068010/o0680102.png" /> of semi-groups with zero, pairwise intersecting at this zero, by specifying on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068010/o0680103.png" /> the multiplication operation that coincides with the original operation on each semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068010/o0680104.png" /> and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068010/o0680105.png" /> for different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068010/o0680106.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068010/o0680107.png" />-direct union is also called the orthogonal sum. A number of types of semi-groups can be described by decomposing them in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068010/o0680108.png" />-direct union of known semi-groups (cf., e.g., [[Maximal ideal|Maximal ideal]]; [[Minimal ideal|Minimal ideal]]; [[Regular semi-group|Regular semi-group]]).
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The [[semi-group]] obtained from the given family $\{S_\alpha\}$ of semi-groups with zero, pairwise intersecting only at the zero element, by specifying on $\bigcup_\alpha S_\alpha$ the multiplication operation that coincides with the original operation on each semi-group $S_\alpha$ and is such that $S_\alpha S_\beta = 0$ for different $\alpha, \beta$. The $O$-direct union is also called the orthogonal sum. A number of types of semi-groups can be described by decomposing them in an $O$-direct union of known semi-groups (cf., e.g., [[Maximal ideal]]; [[Minimal ideal]]; [[Regular semi-group]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''2''' , Amer. Math. Soc.  (1967)</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''2''' , Amer. Math. Soc.  (1967)</TD></TR>
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Revision as of 17:22, 19 June 2016

of semi-groups with zero

The semi-group obtained from the given family $\{S_\alpha\}$ of semi-groups with zero, pairwise intersecting only at the zero element, by specifying on $\bigcup_\alpha S_\alpha$ the multiplication operation that coincides with the original operation on each semi-group $S_\alpha$ and is such that $S_\alpha S_\beta = 0$ for different $\alpha, \beta$. The $O$-direct union is also called the orthogonal sum. A number of types of semi-groups can be described by decomposing them in an $O$-direct union of known semi-groups (cf., e.g., Maximal ideal; Minimal ideal; Regular semi-group).

References

[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 2 , Amer. Math. Soc. (1967)
How to Cite This Entry:
O-direct union. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=O-direct_union&oldid=11404
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article