  Minimal Markov bases of configurations of integer vectors correspond to
minimal binomial generating sets of the assocciated lattice ideal. We give
necessary and sufficient conditions for the elements of a minimal Markov basis
to be (a) inside the universal Gr{\" o}bner basis and (b) inside the Graver
basis. We study properties of Markov bases of generalized Lawrence liftings for
arbitrary matrices $A\in\mathcal{M}_{m\times n}(\Bbb{Z})$ and
$B\in\mathcal{M}_{p\times n}(\Bbb{Z})$ and show that in cases of interest the
{\em complexity} of any two Markov bases is the same.
