Matrices which belong to an idempotent in a sandwich semigroup of circulant Boolean matrices Original Research Article

Let Cn be the semigroup of all the n × n circulant Boolean matrices (n greater-or-equal, slanted 2), and let R be a nonzero element in Cn. The sandwich semigroup of Cn(R) is the set Cn together with the multiplication rule A * B = ARB. For a given idempotent IR in Cn(R), we characterize the matrices A set membership, variant Cn(R) that belong to IR in the sense that A*k = IR for some nonnegative integer k. When R is the identity matrix, the result specializes to a theorem of Schwarz.