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| | ''cyclic semi-group'' | | ''cyclic semi-group'' |
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| − | A [[Semi-group|semi-group]] generated by one element. The monogenic semi-group generated by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m0647301.png" /> is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m0647302.png" /> (sometimes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m0647303.png" />) and consists of all powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m0647304.png" /> with natural exponents. If all these powers are distinct, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m0647305.png" /> is isomorphic to the additive semi-group of natural numbers. Otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m0647306.png" /> is finite, and then the number of elements in it is called the order of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m0647307.png" />, and also the order of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m0647308.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m0647309.png" /> is infinite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473010.png" /> is said to have infinite order. For a finite monogenic semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473011.png" /> there is a smallest number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473012.png" /> with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473013.png" />, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473014.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473015.png" /> is called the index of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473016.png" /> (and also the index of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473017.png" />). In this connection, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473018.png" /> is the smallest number with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473020.png" /> is called the period of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473021.png" /> (of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473022.png" />). The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473023.png" /> is called the type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473024.png" /> (of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473025.png" />). For any natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473027.png" /> there is a monogenic semi-group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473028.png" />; two finite monogenic semi-groups are isomorphic if and only if their types coincide. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473029.png" /> is the type of a monogenic semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473031.png" /> are distinct elements and, consequently, the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473032.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473033.png" />; the set | + | A [[Semi-group|semi-group]] generated by one element. The monogenic semi-group generated by an element $a$ is usually denoted by $\langle a\rangle$ (sometimes by $[a]$) and consists of all powers $a^k$ with natural exponents. If all these powers are distinct, then $\langle a\rangle$ is isomorphic to the additive semi-group of natural numbers. Otherwise $\langle a\rangle$ is finite, and then the number of elements in it is called the order of the semi-group $\langle a\rangle$, and also the order of the element $a$. If $\langle a\rangle$ is infinite, then $a$ is said to have infinite order. For a finite monogenic semi-group $A=\langle a\rangle$ there is a smallest number $h$ with the property $a^h=a^k$, for some $k>h$; $h$ is called the index of the element $a$ (and also the index of the semi-group $A$). In this connection, if $d$ is the smallest number with the property $a^h=a^{h+d}$, then $d$ is called the period of $a$ (of $A$). The pair $(h,d)$ is called the type of $a$ (of $A$). For any natural numbers $h$ and $d$ there is a monogenic semi-group of type $(h,d)$; two finite monogenic semi-groups are isomorphic if and only if their types coincide. If $(h,d)$ is the type of a monogenic semi-group $A=\langle a\rangle$, then $a,\dots,a^{h+d-1}$ are distinct elements and, consequently, the order of $A$ is $h+d-1$; the set |
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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/m/m064/m064730/m06473034.png" /></td> </tr></table>
| + | $$G=\{a^h,\dots,a^{h+d-1}\}$$ |
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| − | is the largest subgroup and smallest ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473035.png" />; the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473036.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473037.png" /> is the unique idempotent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473038.png" />, where <img align= "absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473039.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473041.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473042.png" /> is a [[Cyclic group|cyclic group]], a generator being, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473043.png" />. An idempotent of a monogenic semi-group is a unit (zero) in it if and only if its index (respectively, period) is equal to 1; this is equivalent to the given monogenic semi-group being a [[Group|group]] (respectively, a [[Nilpotent semi-group|nilpotent semi-group]]). Every sub-semi-group of the infinite monogenic semi-group is finitely generated. | + | is the largest subgroup and smallest ideal in $A$; the identity $e$ of the group $G$ is the unique idempotent in $A$, where $e= a^{ld}$ for any $l$ such that $ld\geq h$; $G$ is a [[Cyclic group|cyclic group]], a generator being, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064730/m06473043.png" />. An idempotent of a monogenic semi-group is a unit (zero) in it if and only if its index (respectively, period) is equal to 1; this is equivalent to the given monogenic semi-group being a [[Group|group]] (respectively, a [[Nilpotent semi-group|nilpotent semi-group]]). Every sub-semi-group of the infinite monogenic semi-group is finitely generated. |
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| | ====References==== | | ====References==== |
| | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , '''1''' , Amer. Math. Soc. (1961)</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> | | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , '''1''' , Amer. Math. Soc. (1961)</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> |
Revision as of 08:17, 25 August 2014
cyclic semi-group
A semi-group generated by one element. The monogenic semi-group generated by an element $a$ is usually denoted by $\langle a\rangle$ (sometimes by $[a]$) and consists of all powers $a^k$ with natural exponents. If all these powers are distinct, then $\langle a\rangle$ is isomorphic to the additive semi-group of natural numbers. Otherwise $\langle a\rangle$ is finite, and then the number of elements in it is called the order of the semi-group $\langle a\rangle$, and also the order of the element $a$. If $\langle a\rangle$ is infinite, then $a$ is said to have infinite order. For a finite monogenic semi-group $A=\langle a\rangle$ there is a smallest number $h$ with the property $a^h=a^k$, for some $k>h$; $h$ is called the index of the element $a$ (and also the index of the semi-group $A$). In this connection, if $d$ is the smallest number with the property $a^h=a^{h+d}$, then $d$ is called the period of $a$ (of $A$). The pair $(h,d)$ is called the type of $a$ (of $A$). For any natural numbers $h$ and $d$ there is a monogenic semi-group of type $(h,d)$; two finite monogenic semi-groups are isomorphic if and only if their types coincide. If $(h,d)$ is the type of a monogenic semi-group $A=\langle a\rangle$, then $a,\dots,a^{h+d-1}$ are distinct elements and, consequently, the order of $A$ is $h+d-1$; the set
$$G=\{a^h,\dots,a^{h+d-1}\}$$
is the largest subgroup and smallest ideal in $A$; the identity $e$ of the group $G$ is the unique idempotent in $A$, where $e=a^{ld}$ for any $l$ such that $ld\geq h$; $G$ is a cyclic group, a generator being, for example, 
. An idempotent of a monogenic semi-group is a unit (zero) in it if and only if its index (respectively, period) is equal to 1; this is equivalent to the given monogenic semi-group being a group (respectively, a nilpotent semi-group). Every sub-semi-group of the infinite monogenic semi-group is finitely generated.
References
| [1] | A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , 1 , Amer. Math. Soc. (1961) |
| [2] | E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) |
How to Cite This Entry:
Monogenic semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monogenic_semi-group&oldid=17587
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article