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Let $L$ be a finite graded lattice with rank function $r$. According to the definition on page 39 in the book "Combinatorial Theory" by Aigner, L is said to have the regularity property $(R_t)$ for $t \geq 2$ if for all $x_1,...,x_t \in L$: $$r(x_1 \lor ... \lor x_t)= \sum\limits_{i=1}^{t}{r(x_i)}-\sum\limits_{i<j}^{t}{r(x_i \land x_j)} \pm \cdots (-1)^{t-1} r(x_1 \land \cdots \land x_t).$$

The theorem on page 39 in the book states that $L$ is distributive if and only if $L$ satisfies $(R_3)$ (and then it satisfies all $R_t$ conditions).

Question: Is there a generalisation of this conditions $(R_t)$ for graded posets instead of graded lattices (in the literature, or maybe even an obvious one that I do not see)?

Is there any other place in the literature where those condition $R_t$ appear?

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    $\begingroup$ Not directly related to your question, but is it known if modularity is equivalent to $R_t$ for some $t$? $\endgroup$ Commented Jun 1 at 19:26
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    $\begingroup$ @SamHopkins $R_2$ seems to be equivalent to modular if I understand it correctly on page 39. $\endgroup$ Commented Jun 1 at 19:30

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