complete lattice

Complete lattices

A complete lattice is a poset $P$ such that every subset of $P$ has both a supremum and an infimum in $P$ .

For a complete lattice $L$ ,the supremum of $L$ is denoted by $1$ ,and the infimum of $L$ is denoted by $0$ .Thus $L$ is a bounded lattice,with $1$ as its greatest element and $0$ as its least element.Moreover, $1$ is the infimum of the empty set,and $0$ is the supremum of the empty set.

Generalizations

A countably complete lattice is a poset $P$ such that every countable subset of $P$ has both a supremum and an infimum in $P$ .

Let $\\kappa$ be an infinite cardinal.A $\\kappa$ -complete lattice is a lattice $L$ such that for every subset $A\\subseteq L$ with $|A|\\leq\\kappa$ , both $\\bigvee A$ and $\\bigwedge A$ exist.(Note that an $\\aleph_{0}$ -complete latticeis the same as a countably complete lattice.)

Every complete lattice is a for every infinite cardinal $\\kappa$ ,and in particular is a countably complete lattice.Every countably complete lattice is a bounded lattice.

单词	CompleteLattice
释义	complete lattice Complete lattices A complete lattice is a poset $P$ such that every subset of $P$ has both a supremum and an infimum in $P$ . For a complete lattice $L$ ,the supremum of $L$ is denoted by $1$ ,and the infimum of $L$ is denoted by $0$ .Thus $L$ is a bounded lattice,with $1$ as its greatest element and $0$ as its least element.Moreover, $1$ is the infimum of the empty set,and $0$ is the supremum of the empty set. Generalizations A countably complete lattice is a poset $P$ such that every countable subset of $P$ has both a supremum and an infimum in $P$ . Let $\\kappa$ be an infinite cardinal.A $\\kappa$ -complete lattice is a lattice $L$ such that for every subset $A\\subseteq L$ with $\|A\|\\leq\\kappa$ , both $\\bigvee A$ and $\\bigwedge A$ exist.(Note that an $\\aleph_{0}$ -complete latticeis the same as a countably complete lattice.) Every complete lattice is a for every infinite cardinal $\\kappa$ ,and in particular is a countably complete lattice.Every countably complete lattice is a bounded lattice.
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