Automorphisms and enumeration of switching classes of tournaments.
A completely primary ring is a ring R with identity 1 ≠ 0 whose subset of zero-divisors forms the unique maximal ideal . We determine the structure of the group of automorphisms Aut(R) of a completely primary finite ring R of characteristic p, such that if is the Jacobson radical of R, then ³ = (0), ² ≠ (0), the annihilator of coincides with ² and , the finite field of elements, for any prime p and any positive integer r.
The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.