Pos is the category of small posets and monotone maps.
I call a morphism $f:\mathfrak{A}\rightarrow\mathfrak{B}$ of Pos monovalued iff it maps every atom of $\mathfrak{A}$ either into an atom of $\mathfrak{B}$ or into the least element of $\mathfrak{B}$.
I call a morphism $f:\mathfrak{A}\rightarrow\mathfrak{B}$ of Pos entire iff it maps non-least elements of $\mathfrak{A}$ into non-least elements of $\mathfrak{B}$.
"Monovalued" and "entire" are my terminological inventions. Are there standard terminology for this?