Semimodular lattice

Remarks

Every modular lattice is both upper and lower semimodular, because in a modular lattice the intervals $[a \wedge b, a]$ and $[b, a \vee b]$ are isomorphic.

Example: the lattice of partitions of a finite set, ordered by refinement with the coarser partition above, is upper semimodular. If a partition $a$ covers $a \wedge b$, then $a$ arises from $a \wedge b$ by merging two blocks. Then $a \vee b$ arises from $b$ by merging the two blocks of $b$ that contain these two blocks. These blocks of $b$ are different, since otherwise $a \le b$ and $a = a \wedge b$; so $b \lessdot a \vee b$. For a set of $4$ or more elements, this lattice is not modular.

Non-example: the pentagon lattice N5 with $0 < a < c < 1$ and $b$ incomparable to $a$ and $c$. Apply the condition to the pair $(b, a)$: $b \wedge a = 0$ and $0 \lessdot b$, but $a < c < 1 = a \vee b$, so $a \lessdot a \vee b$ fails.

For a finite lattice, upper semimodularity holds if and only if the lattice is graded with a rank function $r$ satisfying $r(a) + r(b) \ge r(a \wedge b) + r(a \vee b)$ for all $a, b$.