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  "body": "\n## Definition\n\nA *bounded lattice* is a [[lattice|WH4SFAACAS]] $(L, \\le)$ that has a [[greatest element|0XR2QK49V6]] and a [[least element|0AANKKNHS4]].\nThe greatest element is written $1$ or $\\top$, and the least element is written $0$ or $\\bot$.\n\nAlgebraic form: a bounded lattice is a lattice $(L, \\vee, \\wedge)$ with two elements $0, 1 \\in L$ such that $x \\vee 0 = x$ and $x \\wedge 1 = x$ for all $x \\in L$.\n\n## Remarks\n\nThe two forms agree: $x \\vee 0 = x$ for all $x$ says $0 \\le x$ for all $x$, and $x \\wedge 1 = x$ says $x \\le 1$.\nThen also $x \\wedge 0 = 0$ and $x \\vee 1 = 1$.\nExample: the power set of a set $X$, ordered by $\\subseteq$, is a bounded lattice with $0 = \\emptyset$ and $1 = X$.\nEvery subset $A$ satisfies $\\emptyset \\subseteq A \\subseteq X$.\nNon-example: $(\\Z, \\le)$ is a lattice with $x \\vee y = \\max(x, y)$ and $x \\wedge y = \\min(x, y)$, but it has no greatest element.\nEvery finite nonempty lattice $L$ is bounded, with $1 = \\bigvee L$ and $0 = \\bigwedge L$, the join and the meet of all its elements.\n",
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