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  "body": "\n## Definition\n\nLet $(S, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\n$S$ is a *join-semilattice* if every two elements $x, y \\in S$ have a [[join|J2T75ATRV5]] $x \\vee y$ in $S$.\n$S$ is a *meet-semilattice* if every two elements $x, y \\in S$ have a meet $x \\wedge y$ in $S$.\n\nAlgebraic form: a *semilattice* is a set $S$ with a binary operation $\\cdot$ such that, for all $x, y, z \\in S$:\n\n1. $(x \\cdot y) \\cdot z = x \\cdot (y \\cdot z)$;\n2. $x \\cdot y = y \\cdot x$;\n3. $x \\cdot x = x$.\n\n## Remarks\n\nThe two forms are equivalent.\nA join-semilattice gives the operation $x \\cdot y = x \\vee y$.\nConversely, a semilattice $(S, \\cdot)$ becomes a join-semilattice for the order $x \\le y \\iff x \\cdot y = y$, and a meet-semilattice for the order $x \\le y \\iff x \\cdot y = x$.\nSo a semilattice is a commutative [[semigroup|0XMEM406JM]] in which every element is idempotent.\nExample: the finite subsets of $\\N$, ordered by $\\subseteq$, form a join-semilattice, since the union of two finite sets is finite and is their join.\nExample: $\\set{a, b, 1}$ with $a \u003c 1$ and $b \u003c 1$ is a join-semilattice with $a \\vee b = 1$.\nIt is not a meet-semilattice, since $a$ and $b$ have no common lower bound.\nNon-example: a two-element antichain $\\set{a, b}$ is not a join-semilattice, since $a$ and $b$ have no upper bound.\n",
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