Character of a semi-group

semi-group character

A non-zero homomorphism of a commutative semi-group with identity into the multiplicative semi-group consisting of all complex numbers of modulus , 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 . Both concepts of a character of a semi-group are equivalent if is a Clifford semi-group. The set of all characters of a semi-group forms a commutative semi-group with identity (the character semi-group) under pointwise multiplication ,

An ideal of a semi-group is called totally isolated (prime) if 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 . The characters of a commutative semi-group separate the elements of if for any , , there is a such that . If has an identity, then the characters of the semi-group separate the elements of if and only if 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 , , the mapping , , is a character of the semi-group , that is, . The mapping is a homomorphism of into (the so-called canonical homomorphism). If is an isomorphism of onto , then one says that the duality theorem holds for . The duality theorem is true for a commutative semi-group with identity if and only if 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