Locally adequate semigroup algebras

We build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaine idempotents. This allows us to decompose the locally adequate concordant semigroup algebra into a direct product of primitive abundant 0-J*-simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the R*-classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we provide a description of the projective indecomposable modules and determine the representation type.