bounded lattice

A lattice $L$ is said to be if there is an element $0\\in L$ such that $0\\leq x$ for all $x\\in L$ . Dually, $L$ is if there exists an element $1\\in L$ such that $x\\leq 1$ for all $x\\in L$ . A bounded lattice is one that is both from above and below.

For example, any finite lattice $L$ is bounded, as $\\bigvee L$ and $\\bigwedge L$ , being join and meet of finitely many elements, exist. $\\bigvee L=1$ and $\\bigwedge L=0$ .

Remarks.Let $L$ be a bounded lattice with $0$ and $1$ as described above.

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$0\\land x=0$ and $0\\lor x=x$ for all $x\\in L$ .
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$1\\land x=x$ and $1\\lor x=1$ for all $x\\in L$ .
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As a result, $0$ and $1$ , if they exist, are necessarily unique. Forif there is another such a pair $0^{\\prime}$ and $1^{\\prime}$ , then $0=0\\land 0^{\\prime}=0^{\\prime}\\land 0=0^{\\prime}$ . Similarly $1=1^{\\prime}$ .
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$0$ is called the bottom of $L$ and $1$ is called the top of $L$ .
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$L$ is a lattice interval and can be written as $[0,1]$ .

Remark. More generally, a poset $P$ is said to be bounded if it has both a greatest element $1$ and a least element $0$ .

单词	BoundedLattice
释义	bounded lattice A lattice $L$ is said to be if there is an element $0\\in L$ such that $0\\leq x$ for all $x\\in L$ . Dually, $L$ is if there exists an element $1\\in L$ such that $x\\leq 1$ for all $x\\in L$ . A bounded lattice is one that is both from above and below. For example, any finite lattice $L$ is bounded, as $\\bigvee L$ and $\\bigwedge L$ , being join and meet of finitely many elements, exist. $\\bigvee L=1$ and $\\bigwedge L=0$ . Remarks.Let $L$ be a bounded lattice with $0$ and $1$ as described above. • $0\\land x=0$ and $0\\lor x=x$ for all $x\\in L$ . • $1\\land x=x$ and $1\\lor x=1$ for all $x\\in L$ . • As a result, $0$ and $1$ , if they exist, are necessarily unique. Forif there is another such a pair $0^{\\prime}$ and $1^{\\prime}$ , then $0=0\\land 0^{\\prime}=0^{\\prime}\\land 0=0^{\\prime}$ . Similarly $1=1^{\\prime}$ . • $0$ is called the bottom of $L$ and $1$ is called the top of $L$ . • $L$ is a lattice interval and can be written as $[0,1]$ . Remark. More generally, a poset $P$ is said to be bounded if it has both a greatest element $1$ and a least element $0$ .
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