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Difference between revisions of "Regular element"

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''of a semi-group''
 
''of a semi-group''
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r0806901.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r0806902.png" /> for some element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r0806903.png" /> of the given semi-group; if in addition (for the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r0806904.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r0806905.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r0806906.png" /> is called completely regular. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r0806907.png" /> is a regular element of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r0806908.png" />, then the principal right (left) ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r0806909.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r08069010.png" /> is generated by some idempotent; conversely, each of these symmetrical properties implies the regularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r08069011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r08069012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r08069013.png" />, the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r08069014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r08069015.png" /> are called mutually inverse (also known as generalized inverse or regularly conjugate). Every regular element has an element inverse to it; generally speaking, it is not unique (see [[Inversion semi-group|Inversion semi-group]]). Semi-groups in which any two elements are mutually inverse are in fact rectangular semi-groups (see [[Idempotents, semi-group of|Idempotents, semi-group of]]). Each completely-regular element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r08069016.png" /> has an element inverse to it that commutes with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r08069017.png" />. An element is completely regular if and only if it belongs to some subgroup of a semi-group (cf. [[Clifford semi-group|Clifford semi-group]]). For regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080690/r08069018.png" />-classes, see [[Green equivalence relations|Green equivalence relations]].
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An element $a$ such that $a=axa$ for some element $x$ of the given semi-group; if in addition (for the same $x$) $ax=xa$, then $a$ is called completely regular. If $a$ is a regular element of a semi-group $S$, then the principal right (left) ideal in $S$ generated by $a$ is generated by some idempotent; conversely, each of these symmetrical properties implies the regularity of $a$. If $aba=a$ and $bab=b$, the elements $a$ and $b$ are called mutually inverse (also known as generalized inverse or regularly conjugate). Every regular element has an element inverse to it; generally speaking, it is not unique (see [[Inversion semi-group|Inversion semi-group]]). Semi-groups in which any two elements are mutually inverse are in fact rectangular semi-groups (see [[Idempotents, semi-group of|Idempotents, semi-group of]]). Each completely-regular element $a$ has an element inverse to it that commutes with $a$. An element is completely regular if and only if it belongs to some subgroup of a semi-group (cf. [[Clifford semi-group|Clifford semi-group]]). For regular $\mathcal D$-classes, see [[Green equivalence relations|Green equivalence relations]].
  
 
====References====
 
====References====

Latest revision as of 08:24, 12 April 2014

of a semi-group

An element $a$ such that $a=axa$ for some element $x$ of the given semi-group; if in addition (for the same $x$) $ax=xa$, then $a$ is called completely regular. If $a$ is a regular element of a semi-group $S$, then the principal right (left) ideal in $S$ generated by $a$ is generated by some idempotent; conversely, each of these symmetrical properties implies the regularity of $a$. If $aba=a$ and $bab=b$, the elements $a$ and $b$ are called mutually inverse (also known as generalized inverse or regularly conjugate). Every regular element has an element inverse to it; generally speaking, it is not unique (see Inversion semi-group). Semi-groups in which any two elements are mutually inverse are in fact rectangular semi-groups (see Idempotents, semi-group of). Each completely-regular element $a$ has an element inverse to it that commutes with $a$. An element is completely regular if and only if it belongs to some subgroup of a semi-group (cf. Clifford semi-group). For regular $\mathcal D$-classes, see Green equivalence relations.

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)


Comments

A semi-group consisting completely of regular elements is a regular semi-group.

How to Cite This Entry:
Regular element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_element&oldid=31596
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article