  The distance reducing property for Markov bases is an important property that
provides a bound on the mixing time of the associated Markov chain. The goal of
this project is to understand properties of distance-reducing Markov bases. We
explore the distance reducing property for monomial curves and give a complete
characterisation of distance reduction in the case of complete intersection
monomial curves. Our characterisation carefully uses the notion of gluings for
numerical semigroups. We also characterise the distance reducing property for
non-complete intersection monomial curves in small dimensions. We also explore
the distance irreducible elements: the moves that appear in all distance
reducing Markov bases.
