Continuous characters of commutative semigroups



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Let S be a compact commutative topological semigroup and H a closed subsemigroup of S. If [chi] is a continuous unit-character of H, it is possible to obtain the following necessary and sufficient conditions for [chi] to be extendable to S. First, (x,y,a)[epsilon]HxhxS and xa = ya then chi = chi. Also, (x,y)[epsilon]HxH and xe = ye then chi = chi where e is the least idempotent of S. Using these results, if [chi] is a continuous character of S, not necessarily a unit-character, further necessary and sufficient conditions for the extendability of [chi] are found. It is shown that [chi] can be extended to S if and only if there exists an open and closed prime ideal P such that Hintersection is the support of [chi], and if x and y are elements of the support and a an element of the complement of P with xa = ya then chi = chi. From these conditions, other criteria for extendability can be derived with the additional hypothesis that S is a pseudo-invertible semigroup. Finally, results are obtained which show that, to seme extent, it suffices to consider continuous characters defined on closed subsemigroups of S which are unions of components of S. The results in this paper parallel those of R. O. Fulp in his recent paper bearing the same title.