Relatively complemented lattice

Remarks

Equivalently, every interval $[c, d]$ of $L$ is a complemented lattice with least element $c$ and greatest element $d$. A relatively complemented lattice with a least and a greatest element is complemented.

Example: the lattice of all subsets of a set $X$. For $C \subseteq A \subseteq D$, the set $B = C \cup (D \setminus A)$ satisfies $A \cap B = C$ and $A \cup B = D$.

Non-example: the pentagon lattice N5 with $0 < a < c < 1$ and $b$ incomparable to $a$ and $c$. It is complemented, but the interval $[0, c]$ is the chain $0 < a < c$, in which $a$ has no complement.

A complemented modular lattice is relatively complemented: if $x$ is a complement of $a$ and $c \le a \le d$, then $b = c \vee (x \wedge d)$ is a relative complement of $a$ in $[c, d]$.