On the Minimal Majority Operations On a Three-Element Set

Based on a problem originally introduced to the author by Ivo Rosenberg, we give some results about the almost two million clones on a three-element set that contain a (minimal) majority operation. More precisely, for each minimal majority operation m, we determine the least integer k such that every clone containing m is generated by its k-ary part. Moreover, we show that in two of the three essentially different cases for m, Clo(m) is also the largest clone with ternary part Clo(m)((3)), while in one case, it is only the largest clone with 4-ary part Clo(m)((4)). We also discuss some consequences of these results.