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.
Categories: math.AC math.ST stat.TH