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  "body": "\n## Definition\n\nLet $L$ be a [[distributive lattice|0T36P6TXQ6]] with meet $\\wedge$, join $\\vee$ and a least element $0$.\nThe lattice $L$ is a *generalized Boolean algebra* if for all $a, b \\in L$ there is an element $x \\in L$ such that\n$$(a \\wedge b) \\vee x = a \\quad \\text{and} \\quad (a \\wedge b) \\wedge x = 0.$$\nThe element $x$ is unique by distributivity.\nIt is the *relative complement* of $b$ in $a$, written $a \\setminus b$.\n\n## Remarks\n\nExample: the finite subsets of $\\N$, ordered by inclusion, form a generalized Boolean algebra.\nFor finite sets $a$ and $b$, the set $x = a \\setminus b$ is finite, $(a \\cap b) \\cup x = a$, and $(a \\cap b) \\cap x = \\emptyset$.\nThis lattice has no greatest element, so it is not a [[Boolean algebra|06GKBTAA5E]].\nNon-example: the chain $0 \u003c 1 \u003c 2$ is a distributive lattice with least element $0$, but for $a = 2$ and $b = 1$ there is no $x$ with $1 \\vee x = 2$ and $1 \\wedge x = 0$.\nIndeed $1 \\vee x = 2$ forces $x = 2$, and then $1 \\wedge x = 1 \\ne 0$.\nEquivalently, a distributive lattice with $0$ is a generalized Boolean algebra if and only if each interval $[0, a]$ is a Boolean algebra.\nA generalized Boolean algebra with a greatest element is the same as a Boolean algebra.\n",
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