Semigroup algebras of finite ample semigroups. (Q2882515)

Let \(\mathcal R^*\) and \(\mathcal L^*\) denote the generalized Green relations on a semigroup \(S\). Then \(S\) is called an ample semigroup if the set \(E(S)\) of idempotents of \(S\) forms a semilattice, the \(\mathcal R^*\)-class and the \(\mathcal L^*\)-class of every element \(a\in S\) contain exactly one idempotent, denoted by \(a^+\) and \(a^*\), respectively, and \(ae=(ae)^+a\), \(ea=a(ea)^*\). In particular, the class of inverse semigroups coincides with the class of regular ample semigroups. A generalized triangular matrix representation is found for a finite ample semigroup \(S\). A description of the Jacobson radical of the semigroup ring \(A[S]\) over a commutative unital ring \(A\) is derived.