Character of a semi-group

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 Zbl 0118.26501