The Shilov boundary and the Gelfand spectrum of algebras of generalized analytic functions

Let $S$ be discrete abelian semigroup with unit and consellations. We show that the strong boundary and the Shilov boundary of the algebra of generalized analytic functions on the semigroup $\widehat S$ of semicharacters of $S$ are unions of some maximal subgroups of $\widehat S$. If $S$ does not contain nontrivial simple ideals, then both boundaries coincide with the character group of $S$. In this case, the Gelfand spectrum of the algebra under consideration has been calculated.