(edit: I decided to simplify the question and only pose it for bounded posets first)


The Union-closed sets conjecture is equivalent for lattices P to:



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> There exists a join-irreducible element $a$ with $|[a,M]| \leq |P|/2$, when $M$ is the maximum of $P$.
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Recall that an element a of a poset is join-irreducible if there is no subset $X \subseteq P$ with $a\not\in X$ and $a=\bigvee X$.


Call a (finite) bounded poset $P$ lattice-like in case an element $x \in P$ is join-irreducible iff $x$ is covers a unique element.


Every lattice is lattice-like but not every bounded poset is lattice-like.



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> Question 1: Is the above conjecture also true for lattice-like posets?
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This is true for all such posets with at most 8 points. I would think there is a counterexample but I have not found one yet.



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> Question 2: Are there attempts in the literature already to generalise the Union-closed sets conjecture from lattices to a larger class of posets?
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Generalising the union-closed sets conjecture from lattice to a larger class of posets
Here is a counterexample of size 23.


Let $m=6$ and let $$P=\{0,a\_1,\dots,a\_m,1\}\cup\{b\_{ij}: 1\le i<j\le m\}$$
where $0<a\_i<b\_{jk}<1$ whenever $i$ is distinct from $j$ and $k$.


The cardinality of $P$ is $|P|=m+2+\binom{m}{2}=6+2+15=23$.


The join-irreducible elements are only the $a\_i$ and $0$, since
each $b\_{ij}=\bigvee\{a\_k:k\ne i, k\ne j\}$.


Each $|[a\_i,1]|=2+\binom{m}{2}-(m-1)>|P|/2$ as long as
$$2+\binom{m}2 - (m-1) > \frac12\left(m+2+\binom{m}2\right)$$
$$4+\binom{m}2>3m$$
which is true for $m=6$ but not for $m=5$.


