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''on a semi-group''
 
''on a semi-group''
  
Binary relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g0450701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g0450702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g0450703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g0450704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g0450705.png" /> defined as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g0450706.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g0450707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g0450708.png" /> generate identical left principal ideals (cf. [[Principal ideal|Principal ideal]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g0450709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507010.png" /> have a similar meaning after  "left"  has been replaced by  "right"  and  "two-sided" , respectively; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507011.png" /> (union in the lattice of equivalence relations); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507012.png" />. The relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507014.png" /> are commutative in the sense of multiplication of binary relations, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507015.png" /> coincides with their product. The relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507016.png" /> is a right congruence, i.e. is stable from the right: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507017.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507019.png" />; the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507020.png" /> is a left congruence (stable from the left). An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507021.png" />-class and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507022.png" />-class intersect if and only if they are contained in the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507023.png" />-class. All <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507024.png" />-classes in the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507025.png" />-class are equipotent. If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507026.png" />-class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507027.png" /> contains a [[Regular element|regular element]], then all elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507028.png" /> are regular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507029.png" /> contains with some given element all elements inverse to it; such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507030.png" />-class is said to be regular. In a regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507032.png" />-class each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507033.png" />-class and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507034.png" />-class contains an idempotent. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507035.png" /> be an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507036.png" />-class; then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507037.png" /> is a group (which is the case if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507038.png" /> is a maximal subgroup of the given semi-group), or else <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507039.png" />. All group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507040.png" />-classes of the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507041.png" />-class are isomorphic groups. In the general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507042.png" />, but if, for example, some power of each element of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507043.png" /> belongs to a subgroup (in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507044.png" /> is a [[Periodic semi-group|periodic semi-group]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507045.png" />. The inclusion of principal left ideals defines in a natural manner a partial order relation on the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507046.png" />-classes; similar considerations are valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507047.png" />-classes and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045070/g04507048.png" />-classes. These relations were introduced by J. Green [[#References|[1]]].
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Binary relations $\mathcal L$, $\mathcal R$, $\mathcal J$, $\mathcal D$, $\mathcal H$ defined as follows: $x\mathcal Ly$ means that $x$ and $y$ generate identical left principal ideals (cf. [[Principal ideal|Principal ideal]]); $x\mathcal Ry$ and $x\mathcal Jy$ have a similar meaning after  "left"  has been replaced by  "right"  and  "two-sided", respectively; $\mathcal D=L\lor\mathcal R$ (union in the lattice of equivalence relations); $\mathcal H=\mathcal L\cap\mathcal R$. The relations $\mathcal L$ and $\mathcal R$ are commutative in the sense of multiplication of binary relations, so that $\mathcal D$ coincides with their product. The relation $\mathcal L$ is a right congruence, i.e. is stable from the right: $a\mathcal Lb$ implies $ac\mathcal Lbc$ for all $c$; the relation $\mathcal R$ is a left congruence (stable from the left). An $\mathcal L$-class and an $\mathcal R$-class intersect if and only if they are contained in the same $\mathcal D$-class. All $\mathcal H$-classes in the same $\mathcal R$-class are equipotent. If a $\mathcal D$-class $D$ contains a [[Regular element|regular element]], then all elements in $D$ are regular and $D$ contains with some given element all elements inverse to it; such a $\mathcal D$-class is said to be regular. In a regular $\mathcal D$-class each $\mathcal L$-class and each $\mathcal R$-class contains an idempotent. Let $H$ be an arbitrary $\mathcal H$-class; then either $H$ is a group (which is the case if and only if $H$ is a maximal subgroup of the given semi-group), or else $H\cap H^2=\emptyset$. All group $\mathcal H$-classes of the same $\mathcal D$-class are isomorphic groups. In the general case $\mathcal D\neq\mathcal J$, but if, for example, some power of each element of the semi-group $S$ belongs to a subgroup (in particular, if $S$ is a [[Periodic semi-group|periodic semi-group]]), then $\mathcal D=\mathcal J$. The inclusion of principal left ideals defines in a natural manner a partial order relation on the set of $\mathcal L$-classes; similar considerations are valid for $\mathcal R$-classes and $\mathcal J$-classes. These relations were introduced by J. Green [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Green,  "On the structure of semigroups"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 163–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.S. Lyapin,  "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</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"> , ''The algebraic theory of automata, languages and semi-groups'' , Moscow  (1975)  (In Russian; translated from English)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K.H. Hofmann,  P.S. Mostert,  "Elements of compact semigroups" , C.E. Merrill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Green,  "On the structure of semigroups"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 163–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.S. Lyapin,  "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</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"> , ''The algebraic theory of automata, languages and semi-groups'' , Moscow  (1975)  (In Russian; translated from English)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K.H. Hofmann,  P.S. Mostert,  "Elements of compact semigroups" , C.E. Merrill  (1966)</TD></TR></table>

Revision as of 15:48, 4 October 2014

on a semi-group

Binary relations $\mathcal L$, $\mathcal R$, $\mathcal J$, $\mathcal D$, $\mathcal H$ defined as follows: $x\mathcal Ly$ means that $x$ and $y$ generate identical left principal ideals (cf. Principal ideal); $x\mathcal Ry$ and $x\mathcal Jy$ have a similar meaning after "left" has been replaced by "right" and "two-sided", respectively; $\mathcal D=L\lor\mathcal R$ (union in the lattice of equivalence relations); $\mathcal H=\mathcal L\cap\mathcal R$. The relations $\mathcal L$ and $\mathcal R$ are commutative in the sense of multiplication of binary relations, so that $\mathcal D$ coincides with their product. The relation $\mathcal L$ is a right congruence, i.e. is stable from the right: $a\mathcal Lb$ implies $ac\mathcal Lbc$ for all $c$; the relation $\mathcal R$ is a left congruence (stable from the left). An $\mathcal L$-class and an $\mathcal R$-class intersect if and only if they are contained in the same $\mathcal D$-class. All $\mathcal H$-classes in the same $\mathcal R$-class are equipotent. If a $\mathcal D$-class $D$ contains a regular element, then all elements in $D$ are regular and $D$ contains with some given element all elements inverse to it; such a $\mathcal D$-class is said to be regular. In a regular $\mathcal D$-class each $\mathcal L$-class and each $\mathcal R$-class contains an idempotent. Let $H$ be an arbitrary $\mathcal H$-class; then either $H$ is a group (which is the case if and only if $H$ is a maximal subgroup of the given semi-group), or else $H\cap H^2=\emptyset$. All group $\mathcal H$-classes of the same $\mathcal D$-class are isomorphic groups. In the general case $\mathcal D\neq\mathcal J$, but if, for example, some power of each element of the semi-group $S$ belongs to a subgroup (in particular, if $S$ is a periodic semi-group), then $\mathcal D=\mathcal J$. The inclusion of principal left ideals defines in a natural manner a partial order relation on the set of $\mathcal L$-classes; similar considerations are valid for $\mathcal R$-classes and $\mathcal J$-classes. These relations were introduced by J. Green [1].

References

[1] J. Green, "On the structure of semigroups" Ann. of Math. , 54 (1951) pp. 163–172
[2] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[3] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[4] , The algebraic theory of automata, languages and semi-groups , Moscow (1975) (In Russian; translated from English)
[5] K.H. Hofmann, P.S. Mostert, "Elements of compact semigroups" , C.E. Merrill (1966)
How to Cite This Entry:
Green equivalence relations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Green_equivalence_relations&oldid=33489
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article