lattice

lattice

A

lattice

is any poset

L

in which any two

elements

x

and

y

have a least

upper bound

,

x

∨

y

, and a

greatest lower bound

,

x

∧

y

. The

operation

∧

is called

meet

, and the operation

∨

is called

join

. In some literature,

L

is required to be non-empty.

A

sublattice

of

L

is a subposet of

L

which is a lattice, that is, which is closed under the operations

∧

and

∨

as defined in

L

.

The operations of meet and join are

idempotent

,

commutative

,

associative

, and absorptive:

x

∧

(

y

∨

x

)

=

x

⁢

and

⁢

x

∨

(

y

∧

x

)

=

x

.

Thus a lattice is a commutative band with either operation. The

partial order

relation

can be recovered from meet and join by
defining

x

≤

y

⁢

if and only if

⁢

x

∧

y

=

x

.

Once

≤

is defined, it is not hard to see that

x

≤

y

iff

x

∨

y

=

y

as well (one direction goes like:

x

∨

y

=

(

x

∧

y

)

∨

y

=

y

∨

(

x

∧

y

)

=

y

∨

(

y

∧

x

)

=

y

, while the other direction is the dual of the first).

Conspicuously absent from the above list of properties is

distributivity

(

http://planetmath.org/DistributiveLattice

). While many nice lattices, such as face lattices of polytopes, are distributive, there are also important classes of lattices, such as

partition

lattices (

http://planetmath.org/PartitionLattice

), that are usually not distributive.

Lattices, like posets, can be visualized by diagrams called

Hasse diagrams

. Below are two diagrams, both posets. The one on the left is a lattice, while the one on the right is not:

\entrymodifiers

=

[

o

]

⁢

\xymatrix

⁢

@

!

=

1

⁢

p

⁢

t

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

∙

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

∙

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

&

⁢

&

∙

&

⁢

&

\entrymodifiers

=

[

o

]

⁢

\xymatrix

⁢

@

!

=

1

⁢

p

⁢

t

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

∙

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

\ar

⁢

@

-

[

d

]

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

∙

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

∙

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

&

⁢

&

∙

&

⁢

&

The vertices of a lattice diagram can also be labelled, so the lattice diagram looks like

\xymatrix

⁢

@

!

=

1

⁢

p

⁢

t

⁢

&

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

d

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

e

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

f

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

g

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

h

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

i

⁢

&

⁢

&

Remark

. Alternatively, a lattice can be defined as an

algebraic system

. Please see the link below for details.

Title

lattice

\metatable

Generated on Thu Feb 8 20:26:34 2018 by

LaTeXML