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Difference between revisions of "Green equivalence relations"

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''on a semi-group''
 
''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 [[Principal ideal|Principal ideal]]); $x\mathcal Ry$ and $x\mathcal Jy$ have a similar meaning after  "left"  has been replaced by  "right"  and  "two-sided", respectively; $\mathcal D=L\lor\mathcal R$ (union in the lattice of equivalence relations); $\mathcal H=\mathcal L\cap\mathcal R$.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]]), 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]]].
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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; $\mathcal D=\mathcal{L}\lor\mathcal R$ (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]]), 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>
 
<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 {{DOI|10.2307/1969317}} {{ZBL|0043.25601}}</TD></TR>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J. Green, "On the structure of semigroups"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 163–172 {{DOI|10.2307/1969317}} {{ZBL|0043.25601}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian) {{ZBL|0303.20039}}</TD></TR>
+
<TR><TD valign="top">[2]</TD> <TD valign="top">  E.S. Lyapin, "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian) {{ZBL|0303.20039}}</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) {{ZBL|0111.03403}} {{ZBL|0178.01203}}</TD></TR>
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<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) {{ZBL|0111.03403}} {{ZBL|0178.01203}}</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> M.A. Arbib (ed.) ''Algebraic theory of machines, languages and semigroups'' Academic Press (1968) ISBN 0120590506 {{ZBL|0181.01501}} Buslenko, N.P. (ed.), Moscow  (1975)  (In Russian; translated from English)  {{ZBL|0358.94001}}</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top"> M.A. Arbib (ed.) ''Algebraic theory of machines, languages and semigroups'' Academic Press (1968) {{ISBN|0120590506}} {{ZBL|0181.01501}} Buslenko, N.P. (ed.), Moscow  (1975)  (In Russian; translated from English)  {{ZBL|0358.94001}}</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) {{ZBL|0161.01901}}</TD></TR>
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<TR><TD valign="top">[5]</TD> <TD valign="top">  K.H. Hofmann, P.S. Mostert, "Elements of compact semigroups" , C.E. Merrill  (1966) {{ZBL|0161.01901}}</TD></TR>
 
</table>
 
</table>

Latest revision as of 16:59, 25 November 2023

2020 Mathematics Subject Classification: Primary: 20M10 [MSN][ZBL]

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; $\mathcal D=\mathcal{L}\lor\mathcal R$ (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 DOI 10.2307/1969317 Zbl 0043.25601
[2] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) Zbl 0303.20039
[3] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) Zbl 0111.03403 Zbl 0178.01203
[4] M.A. Arbib (ed.) Algebraic theory of machines, languages and semigroups Academic Press (1968) ISBN 0120590506 Zbl 0181.01501 Buslenko, N.P. (ed.), Moscow (1975) (In Russian; translated from English) Zbl 0358.94001
[5] K.H. Hofmann, P.S. Mostert, "Elements of compact semigroups" , C.E. Merrill (1966) Zbl 0161.01901
How to Cite This Entry:
Green equivalence relations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Green_equivalence_relations&oldid=39791
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article