Lattice, structureA
partially ordered set
in which each two-element subset has both a least upper and a greatest
lower bound. This implies the existence of such bounds for every non-empty finite subset.
Examples.
1)
A linearly ordered set (or chain)
where for
 ,
if
 ,
then
2)
The subspaces of a vector space ordered by inclusion, where
3)
The subsets of a given set ordered by inclusion, where
4)
The non-negative integers ordered by divisibility:
if
for a certain
 ;
where
is the least common multiple of
and
 ,
and
is the greatest common divisor of
and
 .
5)
The real-valued functions defined on the interval
and ordered by the condition:
if
for all
 ,
where
in which
and
in which
Let
be a lattice.
becomes a
universal algebra
with two binary operations if one defines
(the symbols
and
or
and
are often used instead of
and
 ).
This universal algebra satisfies the following identities:
Conversely, if
is a set with two binary operations that have the properties
–,
(  )–(  )
mentioned above, then an order
can be imposed on
by setting
if
(it turns out in this case that
if and only if
 ).
The resulting partially ordered set will be a lattice in which
In this way a lattice can be defined as a universal
algebra satisfying the identities
–,
(  )–(  ),
i.e. lattices form a
variety of universal algebras.
If a partially ordered set is regarded as a
small category,
then it is a lattice if and only if has products and coproducts of pairs of objects.
If
and
are lattices and if
 :
is an isomorphism of partially ordered sets, then
is also an isomorphism of the corresponding universal algebras, i.e.
for any
 .
However, an arbitrary
isotone mapping
of
into
is not necessarily a homomorphism of these
lattices considered as universal algebras. Thus, for any
 ,
the mappings
and
are isotone mappings of the lattice
into itself, but they are homomorphisms if and only if
is a
distributive lattice.
However, the first of these mappings is a homomorphism of the semi-lattice
with the operation
 ,
and the second is a homomorphism of the
semi-lattice
with the operation
 .
The class of all lattices forms a category if homomorphisms are taken as morphisms.
An
anti-homomorphism
of a lattice
into a lattice
is a mapping
such that
for any
 .
A composite of two anti-homomorphisms is a homomorphism. A partially ordered
set that is anti-isomorphic to a lattice is a lattice.
By
coordinatization of a lattice
is meant the finding of an
algebraic system
(most often a universal algebra) such that the given lattice
is isomorphic to the lattice of subsystems, to the lattice of
congruences or to some other lattice associated with this algebraic system or
universal algebra. An arbitrary lattice with a 0 and a 1
is coordinatized by the partially ordered semi-group of its residual mappings (cf.
Residual mapping)
into itself, and turns out to be isomorphic to the lattice
of right annihilators of this semi-group. The semi-group itself is a
Baer semi-group,
i.e. the right and left annihilators of each
of its elements are generated by idempotents.
The most important results are obtained for lattices
subjected to some kind of additional restrictions (see
Algebraic lattice;
Atomic lattice;
Brouwer lattice;
Vector lattice;
Modular lattice;
Distributive lattice;
Multiplicative lattice;
Orthomodular lattice;
Complete lattice;
Continuous lattice;
Free lattice;
Lattice with complements;
Boolean algebra).
For specific problems in the theory of lattices see
Ideal;
Filter;
Completion, MacNeille (of a partially ordered set).
Algebraic systems that are at the same time lattices play a special role (see
Lattice-ordered group).
The majority of applications of the theory of lattices are associated with
Boolean algebras. Other classes of lattices have
been used in quantum mechanics and physics.
The concept of a lattice first arose in the late
19th century
and was connected with the fact that many results about the set of
ideals of a ring or the set of normal subgroups of a group
seemed analogous and could be proved in the framework of modular lattices. As
an independent branch of algebra, the theory
of lattices was developed in the
1930s.
References| [1] |
G. Birkhoff,
"Lattice theory"
, Colloq. Publ.
, 25
, Amer. Math. Soc.
(1973) | | [2] |
G. Grätzer,
"General lattice theory"
, Birkhäuser
(1978)
(Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978) | | [3] |
V.N. Salii,
"Lectures on lattice theory"
, Saratov
(1970)
(In Russian) | | [4] |
L.A. Skornyakov,
"Complemented modular lattices and regular rings"
, Oliver & Boyd
(1964)
(Translated from Russian) | | [5] |
L.A. Skornyakov,
"Elements of lattice theory"
, A. Hilger
(1977)
(Translated from Russian) | | [6] |
L.A. Skornyakov,
"Lattice theory"
Itogi Nauk. Algebra, 1964
(1966)
pp. 237–274 | | [7] |
M.M. Glukhov,
I.V. Stelletskii,
T.S. Fofanova,
"Lattice theory"
Progress in Math.
, 12
(1972)
pp. 111–170
Itogi Nauk. Algebra Topol. Geom. 1968
(1970)
pp. 101–154 | | [8] |
, Ordered sets and lattices
, 3 & 7
, Saratov
(1975 & 1983)
(In Russian) | | [9] |
T.S. Blyth,
M.F. Janowitz,
"Residuation theory"
, Pergamon
(1972) | | [10] |
, Ordered sets and lattices (Bratislava, 1985)
, Bratislava
(1989)
(In Russian) | | [11] |
V.N. Salii,
"Lattices with unique complements"
, Moscow
(1984)
(In Russian) | | [12] |
L. Beran,
"Orthomodular lattices"
, Reidel
(1985) | | [13] |
G. Gierz,
K.H. Hofmann,
K. Keimel,
J.D. Lawson,
M.V. Mislove,
D.S. Scott,
"A compendium of continuous lattices"
, Springer
(1980) | | [14] |
G. Kalmbach,
"Orthomodular lattices"
, Acad. Press
(1983) | | [15] |
G. Kalmbach,
"Measures and Hilbert lattices"
, World Sci.
(1986) | | [16] |
R.M. McKenzie,
G.F. McNulty,
R. Taylor,
"Algebras, lattices, varieties"
, 1
, Wadsworth
(1987) | | [17] |
E.T. Schmidt,
"A survey on congruence lattice representations"
, Teubner
(1982) |
L.A. Skornyakov
CommentsNaturally, most theorems in lattice theory require some hypothesis
about the lattice. The remarkable exception is the
Funayama–Nakayama theorem:
The lattice of congruence relations on any lattice is distributive (see e.g.
[1]
or
[2]).
There is also one major unsolved (in
1989)
problem about arbitrary finite
lattices. Every finite lattice is complete and algebraic, and therefore
is representable as the lattice of congruence relations on some
universal algebra
 .
Can
be taken finite?
P.P. Pálfy
and
P. Pudlák
showed
[a4]
that this is closely related to a problem in finite
group theory, which they solved for solvable groups.
W. Feit
[a1]
began the study of the problem in simple groups.
In topology, the awkwardness of Krull dimension (called
in
Dimension
of an associative ring) has been shown to reside only
in the rigidity of the definition. Instead, define the dimension
of a distributive lattice
 ,
like
 ,
as the maximum length of a chain of prime ideals of
 .
Define the dimension
of a topological space
as the minimum of
over lattices of open sets
which form a basis for
 .
Then
for the Noetherian spaces for which
is really used;
for separable metrizable spaces
[a2];
for general metrizable spaces
 ,
 ,
[a3].
The first significant work on lattices was done by
E. Schröder
[a5]
and
R. Dedekind
[a6].
The development of the subject in the
1930-s was largely the work of
G. Birkhoff
[a7]
and
O. Ore
;
the latter used the term
"structure"
instead of
"lattice" ,
but this quickly
became obsolete except in Russia, where it survived until the
1960-s.
References| [a1] |
W. Feit,
"An interval in the subgroup lattice of a finite group which is isomorphic to
 "
Alg. Univ.
, 17
(1983)
pp. 220–221 | | [a2] |
R. Galián,
"Theoriá de la dimensión"
, Madrid
(1979) | | [a3] |
J.R. Isbell,
"Graduation and dimension in locales"
I.H. James (ed.)
E.H. Kronheimer (ed.)
, Aspects of Topology: in Memory of Hugh Dowker
, Lect. notes London Math. Soc.
, 93
, Cambridge Univ. Press
(1985)
pp. 195–210 | | [a4] |
P.P. Pálfy,
P. Pudlák,
"Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups"
Alg. Univ.
, 11
(1980)
pp. 22–27 | | [a5] |
E. Schröder,
"Verlesungen über die Algebra der Logik"
, Teubner
(1890) | | [a6] |
R. Dedekind,
"Ueber die von drei Moduln erzeugte Dualgruppe"
Math. Ann.
, 53
(1900)
pp. 371–403 | | [a7] |
G. Birkhoff,
"On the combination of subalgebras"
Proc. Cambridge Philos. Soc.
, 29
(1933)
pp. 441–464 | | [a8a] |
O. Ore,
"On the foundation of abstract algebra I"
Ann. of Math.
, 36
(1935)
pp. 406–437 | | [a8b] |
O. Ore,
"On the foundation of abstract algebra II"
Ann. of Math.
, 37
(1936)
pp. 265–292 |
This text originally appeared in Encyclopaedia of Mathematics
- ISBN 1402006098
|