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''semi-group character''
 
''semi-group character''
  
A non-zero homomorphism of a commutative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c0215901.png" /> with identity into the multiplicative semi-group consisting of all complex numbers of modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c0215902.png" />, together with 0. Sometimes a character of a semi-group is understood as a non-zero homomorphism into the multiplicative semi-group of complex numbers of modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c0215903.png" />. Both concepts of a character of a semi-group are equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c0215904.png" /> is a [[Clifford semi-group|Clifford semi-group]]. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c0215905.png" /> of all characters of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c0215906.png" /> forms a commutative semi-group with identity (the character semi-group) under [[pointwise multiplication]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c0215907.png" />,
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A non-zero homomorphism of a commutative semi-group $S$ with identity into the multiplicative semi-group consisting of all complex numbers of modulus $1$, together with $0$. Sometimes a character of a semi-group is understood as a non-zero homomorphism into the multiplicative semi-group of complex numbers of modulus $\le 1$. Both concepts of a character of a semi-group are equivalent if $S$ is a [[Clifford semi-group]]. The set $S^*$ of all characters of a semi-group $S$ forms a commutative semi-group with identity (the character semi-group) under [[pointwise multiplication]] ${*}$,
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$$
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(\chi*\psi)(a) = \chi(a)\cdot\psi(a)\,,\ \ a\in S\,,\ \ \chi,\psi\in S^* \ .
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$$
  
<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/c/c021/c021590/c0215908.png" /></td> </tr></table>
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An ideal $P$ of a semi-group $S$ is called ''totally isolated'' (prime) if $S\setminus P$ is a sub-semi-group. The set of all totally-isolated ideals of a commutative semi-group with identity forms a semi-lattice under the operation of union. This semi-lattice is isomorphic to the semi-lattice of idempotents (see [[Idempotents, semi-group of]]) of $S^*$. The characters of a commutative semi-group $S$ separate the elements of $S$ if for any $a,b\in S$, $a\ne b$, there is a $\chi\in S^*$ such that $\chi(a)\ne\chi(b)$. If $S$ has an identity, then the characters of the semi-group $S$ separate the elements of $S$ if and only if $S$ is a [[separable semi-group]]. The problem of describing the character semi-group of an arbitrary commutative semi-group with identity reduces to a description of the characters of a semi-group that is a semi-lattice of groups; for a corresponding description when this semi-lattice satisfies a minimum condition see, for example, [[#References|[1]]]. An abstract characterization of character semi-groups is in [[#References|[2]]].
  
An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c0215909.png" /> of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159010.png" /> is called totally isolated (prime) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159011.png" /> is a sub-semi-group. The set of all totally-isolated ideals of a commutative semi-group with identity forms a semi-lattice under the operation of union. This semi-lattice is isomorphic to the semi-lattice of idempotents (see [[Idempotents, semi-group of|Idempotents, semi-group of]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159012.png" />. The characters of a commutative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159013.png" /> separate the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159014.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159016.png" />, there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159019.png" /> has an identity, then the characters of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159020.png" /> separate the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159021.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159022.png" /> is a [[Separable semi-group|separable semi-group]]. The problem of describing the character semi-group of an arbitrary commutative semi-group with identity reduces to a description of the characters of a semi-group that is a semi-lattice of groups; for a corresponding description when this semi-lattice satisfies a minimum condition see, for example, [[#References|[1]]]. An abstract characterization of character semi-groups is in [[#References|[2]]].
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For every $a\in S$, $\chi\in S^*$, the mapping $\hat a : S^* \rightarrow \mathbf{C}$, $\hat a : \chi \mapsto \chi(a)$, is a character of the semi-group $S^*$, that is, $\hat a \in S^{{*}{*}}$. The mapping $\omega : a \mapsto \hat a$ is a homomorphism of $S$ into $S^{{*}{*}}$ (the so-called canonical homomorphism). If $\omega$ is an isomorphism of $S$ onto $S^{{*}{*}}$, then one says that the duality theorem holds for $S$. The duality theorem is true for a commutative semi-group $S$ with identity if and only if $S$ is an [[inverse semi-group]] [[#References|[3]]]. About duality problems for character semi-groups in the topological case see [[Topological semi-group]].
  
For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159024.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159026.png" />, is a character of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159027.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159028.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159029.png" /> is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159030.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159031.png" /> (the so-called canonical homomorphism). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159032.png" /> is an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159033.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159034.png" />, then one says that the duality theorem holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159035.png" />. The duality theorem is true for a commutative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159036.png" /> with identity if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021590/c02159037.png" /> is an inverse semi-group [[#References|[3]]]. About duality problems for character semi-groups in the topological case see [[Topological semi-group|Topological semi-group]].
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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''' , Amer. Math. Soc. (1961)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  M.M. Lesokhin,  "Characters of commutative semigroups I" ''Izv. Vuz. Mat.'' , '''8'''  (1970) pp. 67–74  (In Russian)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  C. Austin,  "Duality theorems for some commutative semigroups" ''Trans. Amer. Math. Soc.'' , '''109''' : 2  (1963)  pp. 245–256</TD></TR>
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</table>
  
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,  G.B. Preston,  "Algebraic theory of semi-groups" , '''1''' , Amer. Math. Soc.  (1961)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.M. Lesokhin,  "Characters of commutative semigroups I"  ''Izv. Vuz. Mat.'' , '''8'''  (1970)  pp. 67–74  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Austin,  "Duality theorems for some commutative semigroups"  ''Trans. Amer. Math. Soc.'' , '''109''' :  2  (1963)  pp. 245–256</TD></TR></table>
 

Revision as of 19:54, 1 January 2018

semi-group character

A non-zero homomorphism of a commutative semi-group $S$ with identity into the multiplicative semi-group consisting of all complex numbers of modulus $1$, together with $0$. Sometimes a character of a semi-group is understood as a non-zero homomorphism into the multiplicative semi-group of complex numbers of modulus $\le 1$. Both concepts of a character of a semi-group are equivalent if $S$ is a Clifford semi-group. The set $S^*$ of all characters of a semi-group $S$ forms a commutative semi-group with identity (the character semi-group) under pointwise multiplication ${*}$, $$ (\chi*\psi)(a) = \chi(a)\cdot\psi(a)\,,\ \ a\in S\,,\ \ \chi,\psi\in S^* \ . $$

An ideal $P$ of a semi-group $S$ is called totally isolated (prime) if $S\setminus P$ is a sub-semi-group. The set of all totally-isolated ideals of a commutative semi-group with identity forms a semi-lattice under the operation of union. This semi-lattice is isomorphic to the semi-lattice of idempotents (see Idempotents, semi-group of) of $S^*$. The characters of a commutative semi-group $S$ separate the elements of $S$ if for any $a,b\in S$, $a\ne b$, there is a $\chi\in S^*$ such that $\chi(a)\ne\chi(b)$. If $S$ has an identity, then the characters of the semi-group $S$ separate the elements of $S$ if and only if $S$ is a separable semi-group. The problem of describing the character semi-group of an arbitrary commutative semi-group with identity reduces to a description of the characters of a semi-group that is a semi-lattice of groups; for a corresponding description when this semi-lattice satisfies a minimum condition see, for example, [1]. An abstract characterization of character semi-groups is in [2].

For every $a\in S$, $\chi\in S^*$, the mapping $\hat a : S^* \rightarrow \mathbf{C}$, $\hat a : \chi \mapsto \chi(a)$, is a character of the semi-group $S^*$, that is, $\hat a \in S^{{*}{*}}$. The mapping $\omega : a \mapsto \hat a$ is a homomorphism of $S$ into $S^{{*}{*}}$ (the so-called canonical homomorphism). If $\omega$ is an isomorphism of $S$ onto $S^{{*}{*}}$, then one says that the duality theorem holds for $S$. The duality theorem is true for a commutative semi-group $S$ with identity if and only if $S$ is an inverse semi-group [3]. About duality problems for character semi-groups in the topological case see Topological semi-group.

References

[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1 , Amer. Math. Soc. (1961)
[2] M.M. Lesokhin, "Characters of commutative semigroups I" Izv. Vuz. Mat. , 8 (1970) pp. 67–74 (In Russian)
[3] C. Austin, "Duality theorems for some commutative semigroups" Trans. Amer. Math. Soc. , 109 : 2 (1963) pp. 245–256
How to Cite This Entry:
Character of a semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_of_a_semi-group&oldid=35262
This article was adapted from an original article by B.P. TananaL.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article