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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?

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