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Zero morphism $0_{XY}$ is defined by the formulas $a\circ 0_{XY}=b\circ 0_{XY}$ and $0_{XY}\circ c= 0_{XY}\circ d$ for every morphisms $a$, $b$, $c$, $d$ of suitable sources and destinations.

I define a partially ordered category as a category with a partial order on each of its Hom-sets, such that

$$f_1\le f_2\wedge g_1\le g_2 \Rightarrow g_1\circ f_1\le g_2\circ f_2.$$

My question: Is there any connection between zero morphisms and minimal morphisms of a Hom-set? Do they imply each other? Do they imply each other under some additional conditions? Maybe an implication in one or the other direction works?

So: Under which conditions zero morphism and minimal morphism of a Hom-set are the same?

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That these are not equivalent in the general case is clear changing the order of say category $\mathbf{\operatorname{Rel}}$ to the dual order. But they may be equivalent under some additional conditions. – porton Oct 19 '12 at 16:26
No, I don't see any reason why bi-monotone composition should imply that the bottom morphism (assuming it even exists!) is preserved. – Zhen Lin Oct 19 '12 at 17:51

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