Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Lattice (order)
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{short description|Set whose pairs have minima and maxima}} {{distinguish|Lattice (group)}} {{Use shortened footnotes|date=March 2022}} {{More footnotes needed|date=May 2009}} {{stack begin}} {{Binary relations}} {{Algebraic structures |lattice}} {{stack end}} A '''lattice''' is an abstract structure studied in the [[mathematical]] subdisciplines of [[order theory]] and [[abstract algebra]]. It consists of a [[partially ordered set]] in which every pair of elements has a unique [[supremum]] (also called a least upper bound or [[join (mathematics)|join]]) and a unique [[infimum]] (also called a greatest lower bound or [[meet (mathematics)|meet]]). An example is given by the [[power set]] of a set, partially ordered by [[Subset|inclusion]], for which the supremum is the [[Union (set theory)|union]] and the infimum is the [[Intersection (set theory)|intersection]]. Another example is given by the [[natural number]]s, partially ordered by [[divisibility]], for which the supremum is the [[least common multiple]] and the infimum is the [[greatest common divisor]]. Lattices can also be characterized as [[algebraic structure]]s satisfying certain [[axiom]]atic [[Identity (mathematics)|identities]]. Since the two definitions are equivalent, lattice theory draws on both [[order theory]] and [[universal algebra]]. [[Semilattice]]s include lattices, which in turn include [[Heyting algebra|Heyting]] and [[Boolean algebra (structure)|Boolean algebra]]s. These ''lattice-like'' structures all admit [[order-theoretic]] as well as algebraic descriptions. The sub-field of [[abstract algebra]] that studies lattices is called '''lattice theory'''.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)