lattice

The arrangement of atoms within a crystal is often referred to as the crystalline lattice. This is due to the large number of bonds between the atoms (particularly in metallic and ionic materials).

A discrete (additive) subgroup of R^d. A lattice is always isomorphic to Z^d', for some d' < d. This fact is non-trivial to prove, and depends heavily on the discreteness of the lattice.

The form of the isomorphism is also decidedly nontrivial (and nonunique); some very hard problems arise from lattices, e.g. in computational number theory.

A lattice is a partical ordered set (set with a partial order), where every subset containing 2 elements (subsets {a,b}) has a least upper bound (supremum, sup) and a greatest lower bound (infimum, inf).

A lower bound i is a infimum of a set S, iff for every x with for all y of S x <= y , x <= i holds. Dito for supremum with >=.

A lattice is called complete iff every subset has a supremum and an infimum.
A set whose all subsets {a,b} have a supremum is called upper half lattice. In a lower half lattice only the infimums must exist. Upper/lower half lattices are called complete iff the supremums/infimums of all subsets exist.

Examples: Set of subsets of a set is a complete lattice.
Boolean algebras are complete lattices (in fact lattices can be used, along with other constraints, to define a boolean algebra)
The real numbers are a lattice, but not complete.

def.1: a subset of real or complex (or quaternionic) n-space that consists of all finite sums of a set of n independent generating vectors with coefficients in the corresponding ring of integers.

def.2: an algebraic system generalizing the notion of unions and intersections of sets.

--back to combinatorics--

An alternate definition which is equivalent to noaseboar's which seems less arbitrary would be to say that a lattice is any poset where sup H and inf H exist for any finite non-empty subset H of L. This is readily seen to be the same. Let the lattice L be defined as noaseboar has defined it above. We can prove the definitions are identical by induction. Say our subset is H={a}, with only one element. Obviously then, sup H and inf H exist and are both a, because of the reflexive property of our partial ordering relation. Say sup H_n=x and inf H_n=y exist for a subset H_n of L of cardinality n. Let us add one element z to H_n, making H_n+1. Obviously sup H_n+1 exists and is simply sup {x, y}, and inf H_n+1 also exists and is inf {y, z}. So by induction both definitions are equivalent. The infimum or supremum of an empty subset of a lattice need not exist.

Yet another way of looking at lattices is as an algebra <L; ∧ ; ∨> with L a non-void set and ∧ and ∨ are binary operations (known as the meet and join operations respectively) on L which are idempotent, commutative, and associative, and obey the absorption identities:

a ∧ (a ∨ b) = a, and

a ∨ (a ∧ b) = a.

This can easily be shown to be equivalent to noaseboar's definition above by setting sup {a, b} = a ∧ b and inf {a, b} = a ∨ b.

It will be noted that the meet and join operators described above have the same properties as the 'or' and 'and' operators in Boolean algebra.

Lat"tice (?), n. [OE. latis, F. lattis lathwork, fr. latte lath. See Latten, 1st Lath.]

1.

Any work of wood or metal, made by crossing laths, or thin strips, and forming a network; as, the lattice of a window; -- called also latticework.

The mother of Sisera looked out at a window, and cried through the lattice. Judg. v. 28.

2. Her.

The representation of a piece of latticework used as a bearing, the bands being vertical and horizontal.

Lattice bridge, a bridge supported by lattice girders, or latticework trusses. -- Lattice girder Arch., a girder of which the wed consists of diagonal pieces crossing each other in the manner of latticework. -- Lattice plant Bot., an aquatic plant of Madagascar (Ouvirandra fenestralis), whose leaves have interstices between their ribs and cross veins, so as to resemble latticework. A second species is O. Berneriana. The genus is merged in Aponogeton by recent authors.

 

© Webster 1913.

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Lat"tice, v. i. [imp. & p. p. Latticed (?); p. pr. & vb. n. Latticing (?).]

1.

To make a lattice of; as, to lattice timbers.

2.

To close, as an opening, with latticework; to furnish with a lattice; as, to lattice a window.

To lattice up, to cover or inclose with a lattice.

Therein it seemeth he [Alexander] hath latticed up Caesar. Sir T. North.

 

© Webster 1913.