Projective sup-algebras: a general view

We begin with the notion of K-flat projectivity. For each sup-algebra L we then introduce a binary relation ◃L on it. The K-flat projective sup-algebras are exactly such sup-algebras with each element a approximated by the element x, x◃La and the relation ◃L being stable with respect to the operations on L. Further on, we introduce the notion of a K-comonad and characterize K-flat projective sup-algebras as such sup-algebras having a coalgebra structure for the K-comonad.