View source for Bicyclic semi-group

The semi-group with unit element and two generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b0161001.png" /> subject to the single generating relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b0161002.png" />. One of the realizations of the bicyclic semi-group is the Cartesian square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b0161003.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b0161004.png" /> is the set of non-negative integers, with respect to the operation

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The bicyclic semi-group is an [[Inversion semi-group|inversion semi-group]] and as such is monogenic, i.e. is generated by a single element. The idempotents (cf. [[Idempotent|Idempotent]]) of the bicyclic semi-group form a chain, which is ordered with respect to the type of the positive numbers. The bicyclic semi-group is bisimple (cf. [[Simple semi-group|Simple semi-group]]).

The bicyclic semi-group often occurs in theoretical investigations concerning semi-groups, not only as a representative of certain important classes of semi-groups, but also as a  "block" , which defines the structure of individual semi-groups. Thus, for any idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b0161006.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b0161007.png" />-simple, but not completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b0161009.png" />-simple semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610010.png" /> there exists a bicyclic sub-semi-group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610011.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610012.png" /> as the unit element (cf. [[#References|[1]]], Para. 2.7). The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610014.png" /> of the bicyclic semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610015.png" /> defined as above, are, respectively, its left and right multiplying elements (i.e. there exist proper subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610020.png" />). Moreover, in a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610021.png" /> with unit element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610022.png" /> the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610023.png" /> will be a left multiplier if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610024.png" /> contains the bicyclic semi-group whose unit element is identical with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610025.png" />; a similar theorem is also valid for right multiplying elements, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b01610026.png" /> has left multiplying elements if and only if it also has right multiplying elements.

====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>