Editing Lattice (order) (section)

== Examples ==
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Image:Hasse diagram of powerset of 3.svg|'''Pic.&nbsp;1:''' Subsets of <math>\{x, y, z\},</math> under [[set inclusion]]. The name "lattice" is suggested by the form of the [[Hasse diagram]] depicting it.
File:Lattice of the divisibility of 60.svg|'''Pic.&nbsp;2:''' Lattice of integer divisors of 60, ordered by "''divides''".
File:Lattice of partitions of an order 4 set.svg|'''Pic.&nbsp;3:''' Lattice of [[Partition (set theory)|partition]]s of <math>\{1, 2, 3, 4\},</math> ordered by "''refines''".
File:Nat num.svg|'''Pic.&nbsp;4:''' Lattice of positive integers, ordered by <math>\,\leq,</math>
File:N-Quadrat, gedreht.svg|'''Pic.&nbsp;5:''' Lattice of nonnegative integer pairs, ordered componentwise.
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* For any set <math>A,</math> the collection of all subsets of <math>A</math> (called the [[power set]] of <math>A</math>) can be ordered via [[subset inclusion]] to obtain a lattice bounded by <math>A</math> itself and the empty set. In this lattice, the supremum is provided by [[set union]] and the infimum is provided by [[set intersection]] (see Pic.&nbsp;1).
* For any set <math>A,</math> the collection of all finite subsets of <math>A,</math> ordered by inclusion, is also a lattice, and will be bounded if and only if <math>A</math> is finite.
* For any set <math>A,</math> the collection of all [[Partition of a set|partition]]s of <math>A,</math> ordered by [[Partition of a set|refinement]], is a lattice (see Pic.&nbsp;3).
* The [[positive integers]] in their usual order form an unbounded lattice, under the operations of "min" and "max". 1 is bottom; there is no top (see Pic.&nbsp;4).
* The [[Cartesian square]] of the natural numbers, ordered so that <math>(a, b) \leq (c, d)</math> if <math>a \leq c \text{ and } b \leq d.</math> The pair <math>(0, 0)</math> is the bottom element; there is no top (see Pic.&nbsp;5).
* The natural numbers also form a lattice under the operations of taking the [[greatest common divisor]] and [[least common multiple]], with [[divisibility]] as the order relation: <math>a \leq b</math> if <math>a</math> divides <math>b.</math> <math>1</math> is bottom; <math>0</math> is top. Pic.&nbsp;2 shows a finite sublattice.
* Every [[complete lattice]] (also see [[#Completeness|below]]) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical [[Complete lattice#Examples|examples]].
* The set of [[compact element]]s of an [[Arithmetic lattice|arithmetic]] complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property that distinguishes arithmetic lattices from [[algebraic lattice]]s, for which the compacts only form a [[join-semilattice]]. Both of these classes of complete lattices are studied in [[domain theory]].

Further examples of lattices are given for each of the additional properties discussed below.
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