Markov kernels and tribes

We define an ordering on the set of bounded Markov kernels associated with a tribe of fuzzy sets. We show that under this order, the set of bounded Markov kernels is a Dedekind σ-complete lattice. In addition, we define a sum of bounded Markov kernels such that the set of bounded Markov kernels is a lattice-ordered semigroup. If we concentrate only to sharp bounded Markov kernels, then this set is even a Dedekind σ-complete ℓ-group with strong unit. We show that our methods work also for bounded Markov kernels associated with Ts-tribes of fuzzy sets, where Ts is any Frank t-norm and s ∈ (0, ∞).