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Choosing a general set of $n$ points on a curve of genus at least $n$, you can assume that the divisor they define has a unique global section. Let $P$ denote the poset generated by all the possible sums with coefficients $0,1$ of these $n$ points. This poset is the same as the poset of subsets of an $n$-element set.

Since every finite poset seems to be a subposet of the poset of subsets of a finite set [EDIT: this is true-see the comment comments below], just embed your poset in a "power set poset" and remove the unwanted divisors, to deduce that what you want is true.
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Choosing a general set of $n$ points on a curve of genus at least $n$, you can assume that the divisor they define has a unique global section. Let $P$ denote the poset generated by all the possible sums with coefficients $0,1$ of these $n$ points. This poset is the same as the poset of subsets of an $n$-element set.

Since every finite poset seems to be a subposet of the poset of subsets of a finite set [EDIT: this is true-see the comment below], just embed your poset is in a "power set poset" and remove the unwanted divisors, to deduce that what you want is true.
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Choosing a general set of $n$ points on a curve of genus at least $n$, you can assume that the divisor they define has a unique global section. Let $P$ denote the poset generated by all the possible sums with coefficients $0,1$ of these $n$ points. This poset is the same as the poset of subsets of an $n$-element set.

Since every finite poset seems to be a subposet of the poset of subsets of a finite set, just embed your poset is a "power set poset" and remove the unwanted divisors, to deduce that what you want is true.