Matrices which belong to an idempotent in a sandwich semigroup of circulant Boolean matrices (Q2564904)

Let \(C_n\) be the semigroup of all the \(n \times n\) circulant Boolean matrices \((n \geq 2)\), and let \(R\) be a nonzero element in \(C_n\). The sandwich semigroup of \(C_n(R)\) is the set \(C_n\) together with the multiplication rule \(A*B = ARB\). For a given idempotent \(I_R\) in \(C_n(R)\), the author characterizes the matrices \(A \in C_n (R)\) that belong to \(I_R\) in the sense that \(A^{*k} = I_R\) for some nonnegative integer \(k\). When \(R\) is the identity matrix, the result specializes to a theorem of Schwarz.