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text: '[summary: Let $\\langle P, \\leq \\rangle$ be a [3rb poset], and let $S \\subseteq P$. The **join** of $S$ in $P$, denoted by $\\bigvee_P S$, is an element $p \\in P$ satisfying the following two properties:\n\n* p is an *upper bound* of $S$; that is, for all $s \\in S$, $s \\leq p$.\n* For all upper bounds $q$ of $S$ in $P$, $p \\leq q$.\n\n$\\bigvee_P S$ does not necessarily exist, but if it does then it is unique. The notation $\\bigvee S$ is typically used instead of $\\bigvee_P S$ when $P$ is clear from context. Joins are often called *least upper bounds* or *supremums*. For $a, b$ in $P$, the join of $\\{a,b\\}$ in $P$ is denoted by $a \\vee_P b$, or $a \\vee b$ when $P$ is clear from context. **Meets** are greatest lower bounds, and are related to joins by duality. \n]\n\nLet $\\langle P, \\leq \\rangle$ be a [-3rb], and let $S \\subseteq P$. The **join** of $S$ in $P$, denoted by $\\bigvee_P S$, is an element $p \\in P$ satisfying the following two properties:\n\n* p is an *upper bound* of $S$; that is, for all $s \\in S$, $s \\leq p$.\n* For all upper bounds $q$ of $S$ in $P$, $p \\leq q$.\n\n$\\bigvee_P S$ does not necessarily exist, but if it does then it is unique. The notation $\\bigvee S$ is typically used instead of $\\bigvee_P S$ when $P$ is clear from context. Joins are often called *least upper bounds* or *supremums*. For $a, b$ in $P$, the join of $\\{a,b\\}$ in $P$ is denoted by $a \\vee_P b$, or $a \\vee b$ when $P$ is clear from context.\n\nThe dual concept of the join is that of the meet. The **meet** of $S$ in $P$, denoted by $\\bigwedge_P S$, is defined an element $p \\in P$ satisfying.\n\n* p is a *lower bound* of $S$; that is, for all $s$ in $S$, $p \\leq s$.\n* For all lower bounds $q$ of $S$ in $P$, $q \\leq p$.\n\nMeets are also called *infimums*, or *greatest lower bounds*. The notations $\\bigwedge S$, $p \\wedge_P q$, and $p \\wedge q$ are all have meanings that are completely analogous to the aforementioned notations for joins. \n\nBasic example\n--------------------------\n\n\n\nThe above Hasse diagram represents a poset with elements $a$, $b$, $c$, and $d$. $\\bigvee \\{a,b\\}$ does not exist because the set $\\{a,b\\}$ has no upper bounds. $\\bigvee \\{c,d\\}$ does not exist for a different reason: although $\\{c, d\\}$ has upper bounds $a$ and $b$, these upper bounds are incomparable, and so $\\{c, d\\}$ has no *least* upper bound. There do exist subsets of this poset which possess joins; for example, $a \\vee c = a$, $\\bigvee \\{b,c,d\\} = b$, and $\\bigvee \\{c\\} = c$.\n\nNow for some examples of meets. $\\bigwedge \\{a, b, c, d\\}$ does not exist because $c$ and $d$ have no common lower bounds. However, $\\bigwedge \\{a,b,d\\} = d$ and $a \\wedge c = c$.\n\nAdditional Material\n---------------------------------\n\n* [3v4 Examples]\n* [4ll Exercises]\n\nFurther reading\n---------------\n* [46c Lattices]\n',
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