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  "body": "\n## Definition\n\nLet $L$ be a [[lattice|WH4SFAACAS]] with [[join and meet|J2T75ATRV5]] $\\vee$ and $\\wedge$.\nFor $c \\le d$ in $L$, write $[c, d] = \\set{x \\in L : c \\le x \\le d}$.\nA *relative complement* of $a \\in [c, d]$ in $[c, d]$ is an element $b \\in [c, d]$ such that\n$$a \\wedge b = c \\quad\\text{and}\\quad a \\vee b = d.$$\nThe lattice $L$ is *relatively complemented* if for all $c \\le d$ in $L$, every element of $[c, d]$ has a relative complement in $[c, d]$.\n\n## Remarks\n\nEquivalently, every interval $[c, d]$ of $L$ is a [[complemented lattice|ZT47BK9Y6K]] with least element $c$ and greatest element $d$.\nA relatively complemented lattice with a least and a greatest element is complemented.\n\nExample: the lattice of all subsets of a set $X$.\nFor $C \\subseteq A \\subseteq D$, the set $B = C \\cup (D \\setminus A)$ satisfies $A \\cap B = C$ and $A \\cup B = D$.\n\nNon-example: the [[pentagon lattice N5|DHC8TZXGYS]] with $0 \u003c a \u003c c \u003c 1$ and $b$ incomparable to $a$ and $c$.\nIt is complemented, but the interval $[0, c]$ is the chain $0 \u003c a \u003c c$, in which $a$ has no complement.\n\nA complemented [[modular lattice|QX7WZ92QAK]] 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 $[c, d]$.\n",
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