Araújo, JoãoAraújo, João PedroBentz, WolframCameron, PeterSpiga, Pablo2023-01-302023-01-302021-05-01J. Araújo, J.P. Araújo, W. Bentz, P.J. Cameron, and P. Spiga, “A Transversal Property for Permutation Groups Moti- vated by Partial Transformations”, Journal of Algebra 573 (2021), 741-759.http://hdl.handle.net/10400.2/13248Preprint de J. Araújo, J.P. Araújo, W. Bentz, P.J. Cameron, and P. Spiga, “A Transversal Property for Permutation Groups Motivated by Partial Transformations”, Journal of Algebra 573 (2021), 741-759.In this paper we introduce the definition of the (k, l)-universal transversal property for permutation groups, which is a refinement of the definition of k-universal transversal property, which in turn is a refine- ment of the classical definition of k-homogeneity for permutation groups. In particular, a group possesses the (2, n)-universal transversal property if and only if it is primitive; it possesses the (2, 2)-universal transversal property if and only if it is 2-homogeneous. Up to a few undecided cases, we give a classification of groups satisfying the (k, l)-universal transversal property, for k ≥ 3. Then we apply this result for studying regular semigroups of partial transformations.engjournal articlehttps://doi.org/10.1016/j.jalgebra.2020.12.024