View source for Regular element

''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]].

====References====
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR></table>



====Comments====
A semi-group consisting completely of regular elements is a [[Regular semi-group|regular semi-group]].