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''completely-regular semi-group''
 
''completely-regular semi-group''
  
A semi-group in which every element is a group element, that is, lies in some subgroup. An element of a semi-group is a group element if and only if it is completely regular (see [[Regular element|Regular element]]). A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c0224801.png" /> is a Clifford semi-group if and only if either of the following conditions holds: 1) For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c0224802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c0224803.png" />; or 2) each one-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c0224804.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c0224805.png" /> is isolated (or semi-prime), that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c0224806.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c0224807.png" /> for all natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c0224808.png" />.
+
A semi-group in which every element is a group element, that is, lies in some subgroup. An element of a semi-group is a group element if and only if it is completely regular (see [[Regular element|Regular element]]). A semi-group $  S $
 +
is a Clifford semi-group if and only if either of the following conditions holds: 1) For every $  a \in S $,  
 +
$  a \in a  ^ {2} S \cap Sa  ^ {2} $;  
 +
or 2) each one-sided ideal $  I $
 +
of $  S $
 +
is isolated (or semi-prime), that is, if $  x \notin I $
 +
then $  x  ^ {n} \notin I $
 +
for all natural numbers $  n $.
  
Together with inverse semi-groups (cf. [[Inversion semi-group|Inversion semi-group]]), Clifford semi-groups represent one of the most important types of regular semi-groups. Their study was begun in the fundamental paper [[#References|[1]]] of A.H. Clifford. Every Clifford semi-group has a (unique) decomposition into groups, the classes of which are exactly the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c0224809.png" />-classes (cf. [[Green equivalence relations|Green equivalence relations]]). Such a decomposition is not necessarily a band (see [[Band of semi-groups|Band of semi-groups]]); conditions for this to be so are known (see [[#References|[3]]]). The Green relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248011.png" /> on a Clifford semi-group coincide. Every [[Completely-simple semi-group|completely-simple semi-group]] is a Clifford semi-group; a Clifford semi-group is completely simple if and only if it is simple (see [[Simple semi-group|Simple semi-group]]). Every Clifford semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248012.png" /> can be decomposed into a semi-lattice of completely-simple semi-groups; this decomposition is unique, its components are just the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248013.png" />-classes, and the corresponding quotient semi-lattice is isomorphic to the semi-lattice of principal ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248014.png" />. Conversely, every semi-group decomposable into a semi-lattice of completely-simple semi-groups is a Clifford semi-group.
+
Together with inverse semi-groups (cf. [[Inversion semi-group|Inversion semi-group]]), Clifford semi-groups represent one of the most important types of regular semi-groups. Their study was begun in the fundamental paper [[#References|[1]]] of A.H. Clifford. Every Clifford semi-group has a (unique) decomposition into groups, the classes of which are exactly the $  {\mathcal H} $-
 +
classes (cf. [[Green equivalence relations|Green equivalence relations]]). Such a decomposition is not necessarily a band (see [[Band of semi-groups|Band of semi-groups]]); conditions for this to be so are known (see [[#References|[3]]]). The Green relations $  {\mathcal J} $
 +
and $  {\mathcal D} $
 +
on a Clifford semi-group coincide. Every [[Completely-simple semi-group|completely-simple semi-group]] is a Clifford semi-group; a Clifford semi-group is completely simple if and only if it is simple (see [[Simple semi-group|Simple semi-group]]). Every Clifford semi-group $  S $
 +
can be decomposed into a semi-lattice of completely-simple semi-groups; this decomposition is unique, its components are just the $  {\mathcal D} $-
 +
classes, and the corresponding quotient semi-lattice is isomorphic to the semi-lattice of principal ideals of $  S $.  
 +
Conversely, every semi-group decomposable into a semi-lattice of completely-simple semi-groups is a Clifford semi-group.
  
The following conditions are equivalent for a Clifford semi-group: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248015.png" /> is inverse; 2) every idempotent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248016.png" /> lies in the centre, that is, it commutes with every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248017.png" />; 3) every one-sided ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248018.png" /> is two-sided; 4) the Green relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248021.png" /> coincide; 5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248022.png" /> is a semi-lattice of groups; and 6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248023.png" /> is a subdirect product of groups and groups with zero.
+
The following conditions are equivalent for a Clifford semi-group: 1) $  S $
 +
is inverse; 2) every idempotent of $  S $
 +
lies in the centre, that is, it commutes with every element of $  S $;  
 +
3) every one-sided ideal of $  S $
 +
is two-sided; 4) the Green relations $  {\mathcal H} $
 +
and $  {\mathcal D} $
 +
on $  S $
 +
coincide; 5) $  S $
 +
is a semi-lattice of groups; and 6) $  S $
 +
is a subdirect product of groups and groups with zero.
  
The decomposition of an arbitrary Clifford semi-group into a semi-lattice of completely-simple semi-groups defines its  "global structure" . The multiplication law for elements within the components of this decomposition is given by Rees' theorem (see [[Completely-simple semi-group|Completely-simple semi-group]]). Further investigations into Clifford semi-groups are to a large extent directed towards a clarification of their  "fine structure" , that is, to determining multiplication laws of elements from different components. When the components are groups, that is, for inverse Clifford semi-groups, there is a constructive description in terms of a so-called sum of a direct spectrum of groups. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248024.png" /> be a family of pairwise disjoint groups, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248025.png" /> be a semi-lattice (see [[Idempotents, semi-group of|Idempotents, semi-group of]]) such that for every pair of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248027.png" />, there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248029.png" /> is the identity automorphism for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248031.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248032.png" />. A product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248033.png" /> is defined on the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248034.png" /> by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248035.png" /> for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248037.png" />.
+
The decomposition of an arbitrary Clifford semi-group into a semi-lattice of completely-simple semi-groups defines its  "global structure" . The multiplication law for elements within the components of this decomposition is given by Rees' theorem (see [[Completely-simple semi-group|Completely-simple semi-group]]). Further investigations into Clifford semi-groups are to a large extent directed towards a clarification of their  "fine structure" , that is, to determining multiplication laws of elements from different components. When the components are groups, that is, for inverse Clifford semi-groups, there is a constructive description in terms of a so-called sum of a direct spectrum of groups. Let $  \{ G _  \alpha  \} _ {\alpha \in A }  $
 +
be a family of pairwise disjoint groups, let $  A $
 +
be a semi-lattice (see [[Idempotents, semi-group of|Idempotents, semi-group of]]) such that for every pair of elements $  \alpha , \beta \in A $
 +
such that $  \alpha \geq  \beta $,  
 +
there is a homomorphism $  \phi _ {\alpha , \beta }  : G _  \alpha  \rightarrow G _  \beta  $
 +
such that $  \phi _ {\alpha , \alpha }  $
 +
is the identity automorphism for every $  \alpha $
 +
and $  \phi _ {\alpha , \beta }  \circ \phi _ {\beta , \gamma }  = \phi _ {\alpha , \gamma }  $
 +
whenever $  \alpha \geq  \beta \geq  \gamma $.  
 +
A product $  \cdot $
 +
is defined on the union $  S = \cup _ {\alpha \in A }  G _  \alpha  $
 +
by setting $  a \cdot b = a \phi _ {\alpha , \alpha \beta }  b \phi _ {\beta , \alpha \beta }  $
 +
for arbitrary $  a \in G _  \alpha  $
 +
and $  b \in G _  \beta  $.
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022480/c02248038.png" /> becomes an inverse Clifford semi-group. Conversely, every inverse Clifford semi-group can be obtained in this way.
+
Then $  S $
 +
becomes an inverse Clifford semi-group. Conversely, every inverse Clifford semi-group can be obtained in this way.
  
 
The problem of the  "fine structure"  of Clifford semi-groups is in general extremely complicated, and to date (1987) there is no satisfactory solution to it. Certain very complex constructions, describing Clifford semi-groups in terms of completely-simple semi-groups, their translation hulls, semi-lattices, and mappings with special properties, are to be found in [[#References|[5]]]. Great progress has been achieved in the case of orthodox Clifford semi-groups (see [[Regular semi-group|Regular semi-group]]); such semi-groups are called orthogroups. There are a few clear, if somewhat cumbersome, constructions for them (see [[#References|[2]]]). All the constructions mentioned generalize, in some way, the description of inverse Clifford semi-groups obtained in [[#References|[1]]].
 
The problem of the  "fine structure"  of Clifford semi-groups is in general extremely complicated, and to date (1987) there is no satisfactory solution to it. Certain very complex constructions, describing Clifford semi-groups in terms of completely-simple semi-groups, their translation hulls, semi-lattices, and mappings with special properties, are to be found in [[#References|[5]]]. Great progress has been achieved in the case of orthodox Clifford semi-groups (see [[Regular semi-group|Regular semi-group]]); such semi-groups are called orthogroups. There are a few clear, if somewhat cumbersome, constructions for them (see [[#References|[2]]]). All the constructions mentioned generalize, in some way, the description of inverse Clifford semi-groups obtained in [[#References|[1]]].
Line 15: Line 63:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,  "Semigroups admitting relative inverses"  ''Ann. of Math.'' , '''42''' :  4  (1941)  pp. 1037–1049</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.H. Clifford,  "A structure theorem for orthogroups"  ''J. Pure Appl. Algebra'' , '''8''' :  1  (1976)  pp. 23–50</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.S. Lyapin,  "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Petrich,  "The structure of completely regular semigroups"  ''Trans. Amer. Math. Soc.'' , '''189'''  (1974)  pp. 211–236</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,  "Semigroups admitting relative inverses"  ''Ann. of Math.'' , '''42''' :  4  (1941)  pp. 1037–1049</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.H. Clifford,  "A structure theorem for orthogroups"  ''J. Pure Appl. Algebra'' , '''8''' :  1  (1976)  pp. 23–50</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.S. Lyapin,  "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Petrich,  "The structure of completely regular semigroups"  ''Trans. Amer. Math. Soc.'' , '''189'''  (1974)  pp. 211–236</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 17:44, 4 June 2020


completely-regular semi-group

A semi-group in which every element is a group element, that is, lies in some subgroup. An element of a semi-group is a group element if and only if it is completely regular (see Regular element). A semi-group $ S $ is a Clifford semi-group if and only if either of the following conditions holds: 1) For every $ a \in S $, $ a \in a ^ {2} S \cap Sa ^ {2} $; or 2) each one-sided ideal $ I $ of $ S $ is isolated (or semi-prime), that is, if $ x \notin I $ then $ x ^ {n} \notin I $ for all natural numbers $ n $.

Together with inverse semi-groups (cf. Inversion semi-group), Clifford semi-groups represent one of the most important types of regular semi-groups. Their study was begun in the fundamental paper [1] of A.H. Clifford. Every Clifford semi-group has a (unique) decomposition into groups, the classes of which are exactly the $ {\mathcal H} $- classes (cf. Green equivalence relations). Such a decomposition is not necessarily a band (see Band of semi-groups); conditions for this to be so are known (see [3]). The Green relations $ {\mathcal J} $ and $ {\mathcal D} $ on a Clifford semi-group coincide. Every completely-simple semi-group is a Clifford semi-group; a Clifford semi-group is completely simple if and only if it is simple (see Simple semi-group). Every Clifford semi-group $ S $ can be decomposed into a semi-lattice of completely-simple semi-groups; this decomposition is unique, its components are just the $ {\mathcal D} $- classes, and the corresponding quotient semi-lattice is isomorphic to the semi-lattice of principal ideals of $ S $. Conversely, every semi-group decomposable into a semi-lattice of completely-simple semi-groups is a Clifford semi-group.

The following conditions are equivalent for a Clifford semi-group: 1) $ S $ is inverse; 2) every idempotent of $ S $ lies in the centre, that is, it commutes with every element of $ S $; 3) every one-sided ideal of $ S $ is two-sided; 4) the Green relations $ {\mathcal H} $ and $ {\mathcal D} $ on $ S $ coincide; 5) $ S $ is a semi-lattice of groups; and 6) $ S $ is a subdirect product of groups and groups with zero.

The decomposition of an arbitrary Clifford semi-group into a semi-lattice of completely-simple semi-groups defines its "global structure" . The multiplication law for elements within the components of this decomposition is given by Rees' theorem (see Completely-simple semi-group). Further investigations into Clifford semi-groups are to a large extent directed towards a clarification of their "fine structure" , that is, to determining multiplication laws of elements from different components. When the components are groups, that is, for inverse Clifford semi-groups, there is a constructive description in terms of a so-called sum of a direct spectrum of groups. Let $ \{ G _ \alpha \} _ {\alpha \in A } $ be a family of pairwise disjoint groups, let $ A $ be a semi-lattice (see Idempotents, semi-group of) such that for every pair of elements $ \alpha , \beta \in A $ such that $ \alpha \geq \beta $, there is a homomorphism $ \phi _ {\alpha , \beta } : G _ \alpha \rightarrow G _ \beta $ such that $ \phi _ {\alpha , \alpha } $ is the identity automorphism for every $ \alpha $ and $ \phi _ {\alpha , \beta } \circ \phi _ {\beta , \gamma } = \phi _ {\alpha , \gamma } $ whenever $ \alpha \geq \beta \geq \gamma $. A product $ \cdot $ is defined on the union $ S = \cup _ {\alpha \in A } G _ \alpha $ by setting $ a \cdot b = a \phi _ {\alpha , \alpha \beta } b \phi _ {\beta , \alpha \beta } $ for arbitrary $ a \in G _ \alpha $ and $ b \in G _ \beta $.

Then $ S $ becomes an inverse Clifford semi-group. Conversely, every inverse Clifford semi-group can be obtained in this way.

The problem of the "fine structure" of Clifford semi-groups is in general extremely complicated, and to date (1987) there is no satisfactory solution to it. Certain very complex constructions, describing Clifford semi-groups in terms of completely-simple semi-groups, their translation hulls, semi-lattices, and mappings with special properties, are to be found in [5]. Great progress has been achieved in the case of orthodox Clifford semi-groups (see Regular semi-group); such semi-groups are called orthogroups. There are a few clear, if somewhat cumbersome, constructions for them (see [2]). All the constructions mentioned generalize, in some way, the description of inverse Clifford semi-groups obtained in [1].

References

[1] A.H. Clifford, "Semigroups admitting relative inverses" Ann. of Math. , 42 : 4 (1941) pp. 1037–1049
[2] A.H. Clifford, "A structure theorem for orthogroups" J. Pure Appl. Algebra , 8 : 1 (1976) pp. 23–50
[3] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[4] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[5] M. Petrich, "The structure of completely regular semigroups" Trans. Amer. Math. Soc. , 189 (1974) pp. 211–236

Comments

The function symbol is written after the arguments in the article above. This is common practice in the theory of semi-groups.

An extensive bibliography on recent work concerning Clifford semi-groups can be found in [a1] and in the paper of J. Meakin and K.S.S. Nambooripad in [a2].

References

[a1] M. Petrich, "Inverse semigroups" , Wiley (1984)
[a2] G. Pollák (ed.) St. Schwartz (ed.) O. Steinfeld (ed.) , Semigroups , Coll. Math. Soc. J. Bolyai , 39 , North-Holland (1985)
How to Cite This Entry:
Clifford semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clifford_semi-group&oldid=14829
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article