The L-decomposable and the bi-decomposable models are two families of distributions on the set Sn of all permutations of the first n positive integers. Both of these models are characterized by collections of conditional independence relations. We first compute a Markov basis for the L-decomposable model, then give partial results about the Markov basis of the bi-decomposable model. Using these Markov bases, we show that not all bi-decomposable distributions can be approximated arbitrarily well by strictly positive bi-decomposable distributions.
For an arbitrary permutation σ in the semigroup Tn of full transformations on a set with n elements, the regular elements of the centralizer C(σ) of σ in Tn are characterized and criteria are given for C(σ) to be a regular semigroup, an inverse semigroup, and a completely regular semigroup.