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Difference between revisions of "Bicyclic semi-group"

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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 semi-group with unit element $\mathbf{1}$ and two generators $a,b$ subject to the single generating relation $ab = \mathbf{1}$. One of the realizations of the bicyclic semi-group is the Cartesian square $\mathbf{N} \times \mathbf{N}$, where $\mathbf{N}$ is the set of non-negative integers, with respect to the operation
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$$
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(k,l) \cdot (m,n) = (k+m-\min(l,m), l+n - \min(l,m)) \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016100/b0161005.png" /></td> </tr></table>
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The bicyclic semi-group is an [[inversion semi-group]] and as such is [[Monogenic semi-group|monogenic]], i.e. is generated by a single element. The [[idempotent]]s 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]]).
  
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]]).
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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 $e$ of a $0$-simple, but not completely $0$-simple semi-group $S$ there exists a bicyclic sub-semi-group in $S$ containing $e$ as the unit element (cf. [[#References|[1]]], Para. 2.7). The elements $a$ and $b$ of the bicyclic semi-group $B$ defined as above, are, respectively, its left and right multiplying elements (i.e. there exist proper subsets $X$ and $Y$ in $B$ such that $aX = B$, $Yb= B$). Moreover, in a semi-group $S$ with unit element $e$ the element $c$ will be a left multiplier if and only if $S$ contains the bicyclic semi-group whose unit element is identical with $c$; a similar theorem is also valid for right multiplying elements, so that $S$ has left multiplying elements if and only if it also has right multiplying elements.
  
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.
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====References====
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<table>
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<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)  {{ZBL|0111.03403}} {{ZBL|0178.01203}}</TD></TR>
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<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>
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</table>
  
====References====
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<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>
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Latest revision as of 18:22, 16 January 2018

The semi-group with unit element $\mathbf{1}$ and two generators $a,b$ subject to the single generating relation $ab = \mathbf{1}$. One of the realizations of the bicyclic semi-group is the Cartesian square $\mathbf{N} \times \mathbf{N}$, where $\mathbf{N}$ is the set of non-negative integers, with respect to the operation $$ (k,l) \cdot (m,n) = (k+m-\min(l,m), l+n - \min(l,m)) \ . $$

The bicyclic semi-group is an inversion semi-group and as such is monogenic, i.e. is generated by a single element. The idempotents 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).

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 $e$ of a $0$-simple, but not completely $0$-simple semi-group $S$ there exists a bicyclic sub-semi-group in $S$ containing $e$ as the unit element (cf. [1], Para. 2.7). The elements $a$ and $b$ of the bicyclic semi-group $B$ defined as above, are, respectively, its left and right multiplying elements (i.e. there exist proper subsets $X$ and $Y$ in $B$ such that $aX = B$, $Yb= B$). Moreover, in a semi-group $S$ with unit element $e$ the element $c$ will be a left multiplier if and only if $S$ contains the bicyclic semi-group whose unit element is identical with $c$; a similar theorem is also valid for right multiplying elements, so that $S$ has left multiplying elements if and only if it also has right multiplying elements.

References

[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) Zbl 0111.03403 Zbl 0178.01203
[2] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) Zbl 0303.20039
How to Cite This Entry:
Bicyclic semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bicyclic_semi-group&oldid=42741
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article