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  "body": "\n## Definition\n\nLet $(L, \\vee, \\wedge)$ be a [[lattice|WH4SFAACAS]].\nA *sublattice* of $L$ is a nonempty subset $M \\subseteq L$ such that $x \\vee y \\in M$ and $x \\wedge y \\in M$ for all $x, y \\in M$.\n\n## Remarks\n\nA sublattice $M$ is a lattice with the restricted operations, and its order is the order of $L$ restricted to $M$.\nSome authors allow the empty subset as a sublattice.\nExample: in the power set of $\\set{1, 2, 3}$ ordered by $\\subseteq$, the set $M = \\set{\\emptyset, \\set{1}, \\set{2}, \\set{1, 2}}$ is a sublattice.\nThe union and the intersection of two members of $M$ are subsets of $\\set{1, 2}$, and all subsets of $\\set{1, 2}$ are in $M$.\nNon-example: $N = \\set{\\emptyset, \\set{1}, \\set{2}, \\set{1, 2, 3}}$ is a lattice in the induced order, where the join of $\\set{1}$ and $\\set{2}$ is $\\set{1, 2, 3}$.\nIt is not a sublattice, because the join of $\\set{1}$ and $\\set{2}$ in the power set is $\\set{1, 2} \\notin N$.\nEvery chain in a lattice is a sublattice, since $x \\vee y$ and $x \\wedge y$ are $x$ or $y$ when $x$ and $y$ are comparable.\n",
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