Range sets and partition sets in connection with congruences and algebraic invariants

With a semigroup of transformationsS, we associate a class of equivalence relations onR(S) (calledclosed under inclusion relations), the set of ranges ofS. We define a new notion of connectedness for semigroups of transformations (calledrange-connectedness). For a range-connectedS, the closed under inclusion relations and the left-zero congruences ofS are dually isomorphic. The ideas above are dualized for the partition sets ofS. We associate withS an ordered pair which measures its range and partition connectedness. We generalize to an arbitrary semigroupT by considering faithful representations ofT by semigroups of transformations. In so doing, we are able to define an algebraic invariant for semigroups.