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Difference between revisions of "Completely-simple semi-group"

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One of the more important types of simple semi-groups. A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c0240301.png" /> is called completely simple (completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c0240302.png" />-simple) if it is simple (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c0240303.png" />-simple) and contains a primitive idempotent, i.e. a non-zero idempotent that is not an identity for any non-zero idempotent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c0240304.png" />. If a zero is added to a completely-simple semi-group it becomes a completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c0240305.png" />-simple semi-group; for this reason, many properties of completely-simple semi-groups may be deduced directly from the corresponding properties of completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c0240306.png" />-simple semi-groups.
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One of the more important types of simple semi-groups. A semi-group $S$ is called completely simple (completely $0$-simple) if it is simple ($0$-simple) and contains a primitive idempotent, i.e. a non-zero idempotent that is not an identity for any non-zero idempotent of $S$. If a zero is added to a completely-simple semi-group it becomes a completely $0$-simple semi-group; for this reason, many properties of completely-simple semi-groups may be deduced directly from the corresponding properties of completely $0$-simple semi-groups.
  
A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c0240307.png" /> is completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c0240308.png" />-simple if and only if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c0240309.png" />-simple and satisfies one of the following conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403010.png" /> has minimal non-zero left and right ideals; or 2) some power of each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403011.png" /> belongs to a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403012.png" />. In particular, any periodic (and, a fortiori, finite) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403013.png" />-simple semi-group will be a completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403014.png" />-simple semi-group. Any completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403015.png" />-simple semi-group is an O-bisimple [[Regular semi-group|regular semi-group]] and is the union of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403016.png" />-minimal left (right) ideals. A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403017.png" /> is a completely-simple semi-group if and only if it satisfies one of the following conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403018.png" /> is a rectangular band of isomorphic groups (cf. [[Band of semi-groups|Band of semi-groups]]); or 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403019.png" /> is regular and all its idempotents are primitive. A special kind of completely-simple semi-groups is the rectangular group which is the direct product of a group and a rectangular band (cf. [[Idempotents, semi-group of|Idempotents, semi-group of]]). A [[Right group|right group]] (left group) is in turn a special case of a rectangular semi-group. Rees' theorem gives an important representation of completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403020.png" />-simple semi-groups: A semi-group is a completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403021.png" />-simple semi-group if and only if it is isomorphic to a regular [[Rees semi-group of matrix type|Rees semi-group of matrix type]] over a group with zero.
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A semi-group $S$ is completely $0$-simple if and only if it is $0$-simple and satisfies one of the following conditions: 1) $S$ has minimal non-zero left and right ideals; or 2) some power of each element of $S$ belongs to a subgroup of $S$. In particular, any periodic (and, a fortiori, finite) $0$-simple semi-group will be a completely $0$-simple semi-group. Any completely $0$-simple semi-group is an O-bisimple [[Regular semi-group|regular semi-group]] and is the union of its $0$-minimal left (right) ideals. A semi-group $S$ is a completely-simple semi-group if and only if it satisfies one of the following conditions: 1) $S$ is a rectangular band of isomorphic groups (cf. [[Band of semi-groups|Band of semi-groups]]); or 2) $S$ is regular and all its idempotents are primitive. A special kind of completely-simple semi-groups is the rectangular group which is the direct product of a group and a rectangular band (cf. [[Idempotents, semi-group of|Idempotents, semi-group of]]). A [[Right group|right group]] (left group) is in turn a special case of a rectangular semi-group. Rees' theorem gives an important representation of completely $0$-simple semi-groups: A semi-group is a completely $0$-simple semi-group if and only if it is isomorphic to a regular [[Rees semi-group of matrix type|Rees semi-group of matrix type]] over a group with zero.
  
The study of finite completely-simple semi-groups formed the starting point of the development of the theory of semi-groups (cf. [[Semi-group|Semi-group]]). Completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403022.png" />-simple and completely-simple semi-groups frequently appear in various theoretical investigations on semi-groups and are one of the most thoroughly studied types of semi-groups.
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The study of finite completely-simple semi-groups formed the starting point of the development of the theory of semi-groups (cf. [[Semi-group|Semi-group]]). Completely $0$-simple and completely-simple semi-groups frequently appear in various theoretical investigations on semi-groups and are one of the most thoroughly studied types of semi-groups.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403023.png" /> is called simple (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403025.png" />-simple) if it has no proper ideals (respectively, if its only proper ideal is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403027.png" />) (cf. [[Simple semi-group|Simple semi-group]]). More accurately, a primitive idempotent is a non-zero idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403028.png" /> such that for any non-zero idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403030.png" /> only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024030/c02403031.png" /> ( "e is not an identity for any f≠ e" ).
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A semi-group $S$ is called simple ($0$-simple) if it has no proper ideals (respectively, if its only proper ideal is $\{0\}$ and $S^2\neq\{0\}$) (cf. [[Simple semi-group|Simple semi-group]]). More accurately, a primitive idempotent is a non-zero idempotent $e\in S$ such that for any non-zero idempotent $f\in S$, $fe=ef=f$ only if $f=e$ ( "e is not an identity for any f≠ e" ).
  
 
A Rees semi-group of matrix type is often called a Rees matrix semi-group.
 
A Rees semi-group of matrix type is often called a Rees matrix semi-group.

Latest revision as of 20:25, 14 April 2014

One of the more important types of simple semi-groups. A semi-group $S$ is called completely simple (completely $0$-simple) if it is simple ($0$-simple) and contains a primitive idempotent, i.e. a non-zero idempotent that is not an identity for any non-zero idempotent of $S$. If a zero is added to a completely-simple semi-group it becomes a completely $0$-simple semi-group; for this reason, many properties of completely-simple semi-groups may be deduced directly from the corresponding properties of completely $0$-simple semi-groups.

A semi-group $S$ is completely $0$-simple if and only if it is $0$-simple and satisfies one of the following conditions: 1) $S$ has minimal non-zero left and right ideals; or 2) some power of each element of $S$ belongs to a subgroup of $S$. In particular, any periodic (and, a fortiori, finite) $0$-simple semi-group will be a completely $0$-simple semi-group. Any completely $0$-simple semi-group is an O-bisimple regular semi-group and is the union of its $0$-minimal left (right) ideals. A semi-group $S$ is a completely-simple semi-group if and only if it satisfies one of the following conditions: 1) $S$ is a rectangular band of isomorphic groups (cf. Band of semi-groups); or 2) $S$ is regular and all its idempotents are primitive. A special kind of completely-simple semi-groups is the rectangular group which is the direct product of a group and a rectangular band (cf. Idempotents, semi-group of). A right group (left group) is in turn a special case of a rectangular semi-group. Rees' theorem gives an important representation of completely $0$-simple semi-groups: A semi-group is a completely $0$-simple semi-group if and only if it is isomorphic to a regular Rees semi-group of matrix type over a group with zero.

The study of finite completely-simple semi-groups formed the starting point of the development of the theory of semi-groups (cf. Semi-group). Completely $0$-simple and completely-simple semi-groups frequently appear in various theoretical investigations on semi-groups and are one of the most thoroughly studied types of semi-groups.

References

[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[2] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[3] K. Kapp, H. Schneider, "Completely 0-simple semigroups: an abstract treatment of the lattice of congruences" , Benjamin (1969)


Comments

A semi-group $S$ is called simple ($0$-simple) if it has no proper ideals (respectively, if its only proper ideal is $\{0\}$ and $S^2\neq\{0\}$) (cf. Simple semi-group). More accurately, a primitive idempotent is a non-zero idempotent $e\in S$ such that for any non-zero idempotent $f\in S$, $fe=ef=f$ only if $f=e$ ( "e is not an identity for any f≠ e" ).

A Rees semi-group of matrix type is often called a Rees matrix semi-group.

References

[a1] A.K. Suschkewitsch, "Ueber die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" Math. Ann. , 99 (1928) pp. 30–50
[a2] D. Rees, "On semi-groups" Proc. Cambridge Phil. Soc. , 36 (1940) pp. 387–400
How to Cite This Entry:
Completely-simple semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-simple_semi-group&oldid=18594
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article