Coregular semigroups of full transformations

This paper is mainly dedicated to the description of coregular subsemigroups of the symmetric semigroup of transformations on an -element set. Namely, we characterize all coregular transformation semigroups with | | ≤ 3. In the subsemigroup of all extensive transformations, the coregular elements coincide with the idempotent ones. We characterize all bands within . Within the subsemigroup of all order-preserving extensive transformations, we also determine the maximal bands (with respect to the inclusion).