Orderly Majority Functions with their Relation to Minimal Clones

For majority functions on finite sets, we define the concepts of orderly and doubly orderly and prove the following. For a 2-valued (on triples with mutually distinct components) majority function
$f$
, the clone generated by
$f$
always contains an orderly majority function. With an additional assumption imposed on some values of
$f$
, the clone generated by
$f$
contains a doubly orderly majority function. Based on these observations, we present a new class of minimal functions.