## Lattice (order) |

This article includes a its sources remain unclear because it has insufficient . (May 2009) ( |

A **lattice** is an abstract structure studied in the

Lattices can also be characterized as

- lattices as partially ordered sets
- lattices as algebraic structures
- connection between the two definitions
- examples
- counter-examples
- morphisms of lattices
- sublattices
- properties of lattices
- free lattices
- important lattice-theoretic notions
- see also
- notes
- references
- external links

If (*L*, ≤) is a *S* ⊆ *L* is an arbitrary subset, then an element *u* ∈ *L* is said to be an **upper bound** of *S* if *s* ≤ *u* for each *s* ∈ *S*. A set may have many upper bounds, or none at all. An upper bound *u* of *S* is said to be its **least upper bound**, or ** join**, or

A partially ordered set (*L*, ≤) is called a ** join-semilattice** if each two-element subset {

It follows by an *see*

A **bounded lattice** is a lattice that additionally has a ** greatest** element 1 and a

- 0 ≤
*x*≤ 1 for every*x*in*L*.

The greatest and least element is also called the **maximum** and **minimum**, or the **top** and **bottom** element, and denoted by ⊤ and ⊥, respectively. Every lattice can be converted into a bounded lattice by adding an artificial greatest and least element, and every non-empty finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by (resp.) where .

A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. For every element *x* of a poset it is trivially true (it is a *A* and *B* of a poset *L*,

and

hold. Taking *B* to be the empty set,

and

which is consistent with the fact that .

A lattice element *y* is said to ** cover** another element

A lattice (*L*, ≤) is called ** graded**, sometimes

Given a subset of a lattice, *H* ⊂ *L*, meet and join restrict to *H*. The resulting structure on *H* is called a **partial lattice**. In addition to this extrinsic definition as a subset of some other algebraic structure (a lattice), a partial lattice can also be intrinsically defined as a set with two partial binary operations satisfying certain axioms.^{[1]}

Other Languages

العربية: شبكية (ترتيب)

català: Reticle (ordre)

čeština: Svaz (matematika)

dansk: Gitter (ordning)

Deutsch: Verband (Mathematik)

eesti: Võre (matemaatika)

español: Retículo (matemáticas)

Esperanto: Latiso (matematiko)

فارسی: مشبکه (ترتیب)

français: Treillis (ensemble ordonné)

한국어: 격자 (순서론)

italiano: Reticolo (matematica)

עברית: סריג (מבנה סדור)

magyar: Háló (matematika)

Nederlands: Tralie (wiskunde)

日本語: 束 (束論)

Piemontèis: Retìcol

polski: Krata (porządek)

português: Reticulado

русский: Решётка (алгебра)

slovenčina: Zväz (matematika)

suomi: Hila (matematiikka)

svenska: Gitter (ordning)

українська: Ґратка (порядок)

中文: 格 (数学)