# Generalising the union-closed sets conjecture from lattice to a larger class of posets

(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:

There exists a join-irreducible element $$a$$ with $$|[a,M]| \leq |P|/2$$, when $$M$$ is the maximum of $$P$$.

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.

Question 1: Is the above conjecture also true for lattice-like posets?

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.

Question 2: Are there attempts in the literature already to generalise the Union-closed sets conjecture from lattices to a larger class of posets?

Let $$m=6$$ and let $$P=\{0,a_1,\dots,a_m,1\}\cup\{b_{ij}: 1\le i where $$0 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 $$m=5$$.