Difference between revisions of "Arguesian lattice"
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See [[#References|[a16]]]. Every modular lattice with complements (cf. [[Lattice with complements|Lattice with complements]]) can be embedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068023.png" /> for a projective space on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068024.png" /> of its maximal filters (cf. [[Filter|Filter]]), actually a sublattice of the ideal lattice of the filter lattice (with filters ordered by inverse inclusion), whence preserving all identities. This Frink embedding generalizes the Stone representation theorem for Boolean algebras (cf. [[Boolean algebra|Boolean algebra]]). The coordinatization theorem of [[Projective geometry|projective geometry]] implies that any Arguesian relatively complemented lattice can be embedded into a [[Direct product|direct product]] of lattices of subspaces of vector spaces (cf. [[Vector space|Vector space]]) [[#References|[a22]]]. | See [[#References|[a16]]]. Every modular lattice with complements (cf. [[Lattice with complements|Lattice with complements]]) can be embedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068023.png" /> for a projective space on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068024.png" /> of its maximal filters (cf. [[Filter|Filter]]), actually a sublattice of the ideal lattice of the filter lattice (with filters ordered by inverse inclusion), whence preserving all identities. This Frink embedding generalizes the Stone representation theorem for Boolean algebras (cf. [[Boolean algebra|Boolean algebra]]). The coordinatization theorem of [[Projective geometry|projective geometry]] implies that any Arguesian relatively complemented lattice can be embedded into a [[Direct product|direct product]] of lattices of subspaces of vector spaces (cf. [[Vector space|Vector space]]) [[#References|[a22]]]. | ||
− | A compact element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068025.png" /> of a modular [[Algebraic lattice|algebraic lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068026.png" /> is called a point if it is | + | A compact element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068025.png" /> of a modular [[Algebraic lattice|algebraic lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068026.png" /> is called a point if it is a [[join-irreducible element]], i.e. has a unique lower cover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068027.png" />. If each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068028.png" /> is a join of points (e.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068029.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068030.png" /> can be understood as the subspace lattice of an ordered linear space on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068031.png" /> of points: the order is induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068032.png" />. Points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068035.png" /> are collinear if they are distinct and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068036.png" />, and a subspace is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068037.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068038.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068039.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068040.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068043.png" /> collinear implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068044.png" />. This can also be viewed as a presentation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068045.png" /> as a [[Semi-lattice|semi-lattice]]. Instead of all collinearities one may use a base of lines: for each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068046.png" /> a maximal set of points with pairwise join <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068047.png" />. For an abstract ordered linear space one has to require that collinearity is a totally symmetric relation, that collinear points are incomparable, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068051.png" /> collinear implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068052.png" />, that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068056.png" /> collinear there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068058.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068061.png" /> are collinear or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068062.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068063.png" />, and, finally, a more elaborate version of the triangle axiom. Then the subspaces form a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110680/a11068064.png" /> as above and each modular lattice can be naturally embedded into such, preserving identities. |
==Subdirect products and congruences.== | ==Subdirect products and congruences.== |
Revision as of 18:52, 12 June 2016
Desarguesian lattice
A lattice in which the Arguesian law is valid, i.e. for all ,
,
![]() |
,
for any permutation
[a21]. Arguesian lattices form a variety (cf. also Algebraic systems, variety of), since within lattices
is equivalent to
. A lattice is Arguesian if and only if it is a modular lattice and
(central perspectivity) implies
(axial perspectivity). In an Arguesian lattice and for
,
such that
and
, the converse implication is valid too [a24]. A lattice is Arguesian if and only if its partial order dual is Arguesian.
Examples of Arguesian lattices.
1) The lattice of subspaces of a projective space
is Arguesian if and only if the Desargues assumption is satisfied in
.
2) Every lattice of submodules of an
-module
(cf. also Module) and any lattice of subobjects of an object in an Abelian category.
3) Every lattice of normal subgroups (respectively, congruence relations; cf. Normal subgroup; Congruence (in algebra)) of a group and any lattice of permuting equivalence relations [a21] (also called a linear lattice).
4) Considering all lattices of congruence relations of algebraic systems (cf. Algebraic system) in a variety, the Arguesian law is equivalent to the modular law.
5) Every -distributive modular lattice (cf. also Distributive lattice):
, i.e. without a projective plane in the variety.
The Arguesian law can be characterized in terms of forbidden subconfigurations, but not in terms of sublattices [a17]. Weaker versions involve less variables and higher-dimensional versions have increasing strength and number of variables; all are valid in linear lattices [a10]. The basic structure theory relies on the modular law, cf. Modular lattice and [a3], [a27]. For its role in the congruence and commutator theory of algebraic systems, cf. [a12]. Large parts of dimension theory for rings and modules can be conveniently done within modular lattices [a29].
Projective spaces.
See [a16]. Every modular lattice with complements (cf. Lattice with complements) can be embedded into for a projective space on the set
of its maximal filters (cf. Filter), actually a sublattice of the ideal lattice of the filter lattice (with filters ordered by inverse inclusion), whence preserving all identities. This Frink embedding generalizes the Stone representation theorem for Boolean algebras (cf. Boolean algebra). The coordinatization theorem of projective geometry implies that any Arguesian relatively complemented lattice can be embedded into a direct product of lattices of subspaces of vector spaces (cf. Vector space) [a22].
A compact element of a modular algebraic lattice
is called a point if it is a join-irreducible element, i.e. has a unique lower cover
. If each element of
is a join of points (e.g., if
), then
can be understood as the subspace lattice of an ordered linear space on the set
of points: the order is induced by
. Points
,
,
are collinear if they are distinct and
, and a subspace is a subset
such that
implies
, and
with
,
,
collinear implies
. This can also be viewed as a presentation of
as a semi-lattice. Instead of all collinearities one may use a base of lines: for each element
a maximal set of points with pairwise join
. For an abstract ordered linear space one has to require that collinearity is a totally symmetric relation, that collinear points are incomparable, that
and
,
,
collinear implies
, that for
and
,
,
collinear there are
and
such that
,
,
are collinear or
or
, and, finally, a more elaborate version of the triangle axiom. Then the subspaces form a lattice
as above and each modular lattice can be naturally embedded into such, preserving identities.
Subdirect products and congruences.
See [a3], [a20]. Every lattice is a subdirect product of subdirectly irreducible homomorphic images (cf. Homomorphism). By Jónsson's lemma, the subdirect irreducibles in the variety generated by a class are homomorphic images of sublattices of ultraproducts from
. A pair of complementary central elements
,
provides a direct decomposition
, a neutral element
implies a subdirect decomposition
.
Any congruence on a modular lattice
is determined by its set
of quotients, where a quotient is a pair
with
, equivalently, an interval
. A pair of quotients is projective if it belongs to the equivalence relation generated by
,
such that
and
. A subquotient
of
is such that
. If
is generated by a set
of quotients, then
is the transitive closure of the set of all quotients projective to some subquotient of a quotient in
. The congruences form a Brouwer lattice, with the pseudo-complement
of
given by the quotients not having any subquotient projective to a subquotient of a quotient in
.
is subdirectly decomposed into
and
and each subdirectly indecomposable factor of
is a homomorphic image of
or
. If
is onto,
, and if
(which then preserves sups) and the dual
exist, i.e. for a bounded image, then for
one finds that
is the transitive closure of prime quotients
with
,
for some prime quotient
in
. For any onto mapping
with
not factoring through
, this splitting method yields the relations
for prime quotients
in
. If
is generated by a finite set
, starting with
and iterating,
with
,
,
ranging over all subtriples of lines of a given base, leads to
for some
[a28].
For , each congruence is determined by its prime quotients, either those in a given composition sequence or those of the form
,
a point. It follows that the congruences form a finite Boolean algebra and are in one-to-one correspondence with unions of connected components of the point set under the binary relation:
with
,
,
collinear. Moreover, the subdirectly indecomposable factors
of
are simple, i.e. correspond to maximal congruences
, and the dimensions add up:
. The connected components associated with the
are disjoint and are isomorphic images of the spaces of the
via
. Thus, the space of
can be constructed as the disjoint union of the spaces of the
with
if and only if
where
depends only on the subdirect product of
and
and can be computed, in the scaffolding construction, as the pointwise largest sup-homomorphism
of
into
such that
for a given set of generators
.
Glueing.
See [a8]. A tolerance relation on a lattice is a binary relation that is reflexive, symmetric, and compatible, i.e. a subalgebra of
. A block is a maximal subset with every pair of elements in relation, whence a convex sublattice. The set
of blocks has a lattice structure. A convenient way to think of this is as a pair
of embeddings of a (not necessarily modular) skeleton lattice
into the filter, respectively ideal, lattice of
preserving finite sups, respectively infs, such that
is non-empty for each
, namely one of the blocks. A relevant tolerance for modular lattices is given by the relation that
be complemented. Its blocks are the maximal relatively complemented convex sublattices of
, and
is then the prime skeleton. One has a glueing if the smallest congruence extending the tolerance is total; this occurs for modular
of
and the prime skeleton tolerance. The neutrality of
can be shown with suitable
via an order-preserving mapping
turning
into a glueing with blocks
,
; this happens if:
is sup-preserving,
is inf-preserving, and for each
in some generating set there is an
with
.
Every lattice with a tolerance gives rise to a system of adjunctions between the blocks
,
,
in
, satisfying certain axioms. Namely,
if and only if
if and only if
. Conversely, each such system defines a pre-order on the disjoint union of the
and, factoring by the associated equivalence relation, a lattice with tolerance having blocks
. Glueing always produces a modular lattice from modular blocks, but only in special cases the impact of the Arguesian law and various kinds of representability are understood (a necessary condition is that any pair of adjunctions matching coordinate rings of two frames induces an anti-isomorphism of partially ordered sets [a17]). For the combinatorial analysis of subgroup lattices of finite Abelian groups, cf. [a2].
Coordinates.
See [a5], [a7]. J. von Neumann introduced the lattice-theoretic analogue of projective coordinate systems: an -frame consists of independent elements
,
,
,
, such that
,
,
, and
. There are equivalent variants. Any
provides frames
,
and
,
, where
, of sublattices which can be used to derive frames satisfying relations. The elements
such that
and
form the coordinate domain
. For a free
-module with basis
one has the canonical frame
,
and
. If
or, in the presence of the Arguesian law,
[a6], then the
are turned into rings (cf. Ring) isomorphic via
, respectively
, with unit
and
![]() |
![]() |
Every modular lattice generated by a frame can be generated by elements. Every finitely-generated semi-group
can be embedded into the multiplicative semi-group of the coordinate ring of a suitable frame in some
-generated sublattice of
over a given field
(finite dimensional if
is finite).
A complemented Arguesian lattice possessing a large partial -frame (i.e., a
-frame of a section
with
having a complement
,
perspective to
) or being simple of dimension
is isomorphic to the lattice of principal right ideals of some regular ring [a23]. Under suitable richness assumptions, lattices
have been characterized for various classes of rings via the Arguesian law and geometric conditions on the lattice, e.g. for completely primary uniserial rings [a24] and left Ore domains. There are results on lattice isomorphisms induced by semi-linear mappings, respectively Morita equivalences (cf. also Morita equivalence), and on lattice homomorphisms induced by tensoring [a1]. Abelian lattices, having certain features of Abelian categories, can be embedded into subgroup lattices of Abelian groups. This includes algebraic modular lattices having an infinite frame [a32].
Equational theory.
See [a5], [a7], [a8], [a20]. The class of all linear lattices, respectively the class of all lattices embeddable into some
, forms a quasi-variety, since it arises from a projective class in the sense of Mal'tsev. Natural axiom systems and proof theories for quasi-identities have been given, cf. [a10], [a33]. The latter present identities via graphs. On the other hand, there is no finitely-axiomatized quasi-variety containing
,
some field, and satisfying all higher-dimensional Arguesian laws. Also, every quasi-variety of modular lattices containing some
also contains a
-generated finitely-presented lattice with unsolvable decision problem for words [a18].
Identities are preserved when passing to the ideal lattice; thus, one may assume algebraicity. Frames are projective systems of generators and relations within modular lattices: for each there are terms
,
in the variables
,
such that the
,
form a frame in a sublattice for any choice of the
,
in a modular lattice and
,
if these happen to form a frame already. This allows one to translate divisibility of integer multiples of
in a ring (more generally, solvability of systems of linear equations with integer coefficients) into lattice identities. The converse has been done in [a19] for lattices of submodules: solving the decision problem for words in free lattices in
, whenever
has decidable divisibility of integers (e.g.
), and providing a complete list of all varieties
, each generated by finite-dimensional members (related ideas occur in the model theory of modules [a31]). In contrast, no finitely-axiomatized variety of modular lattices containing
is generated by its finite-dimensional members. For free lattices with
generators in the quasi-varieties of all Arguesian linear, respectively normal, subgroup lattices the decision problem remains open (in contrast to the negative answer for modular lattices [a11]). The corresponding variety containments, with
included, are all proper [a25], [a26], [a30]. There are rings
with
not a variety, but the status for
,
a field,
, normal subgroup and linear lattices is unknown. Yet, for finite-dimensional
a retraction into
is possible. The variety generated by modular lattices of
can be finitely axiomatized; for
the lattice of subvarieties and the covering varieties have been determined [a20]. Finitely-generated varieties are finitely axiomatizable (this does not extend to quasi-varieties).
Generators and relations.
See [a28]. Given a pair ,
of complements in a modular lattice
and a subset
such that
for all
, one has that
,
are central in the sublattice they generate together with
. This applies to a direct decomposition
of a representation of a partially ordered set,
, with
. Hence, for a set
of generators with partial order relation, the subdirectly indecomposable factors of the free lattice in
can be obtained via Jónsson's lemma from the subdirectly indecomposable factors of indecomposable finite-dimensional representations. In particular, this carries through for representation-finite
. For
not containing
nor
, these are exactly the subdirectly indecomposable modular lattices generated by such
, namely
- or
-element. For
one obtains all
,
,
the prime subfield, lattices with
, and a series of
-distributives (with
labelings by generators) [a13]. The latter are exactly the subdirectly indecomposable modular lattices generated by two pairs of complements. Also, the structure of the free lattices in
over these and other tame
of finite growth is understood [a4]. Moreover, the word problem for
-generated finitely-presented lattices in
is solvable. The lattice-theoretic approach determines the subdirectly indecomposable factors
, first, using neutral elements and the splitting method.
A large number of finitely-presented modular lattices with additional unary operations have been determined in [a14], [a28] as invariants for the orbits (cf. Orbit) of subspaces under the group of isometric mappings (cf. Isometric mapping) of a vector space endowed with a sesquilinear form. The above methods have been modified to this setting.
The Arguesian lattices generated by a frame can be explicitly determined as certain lattices of subgroups of Abelian groups. To some extent the analysis for and other generating posets carries over to Arguesian lattices, but essentially new phenomena occur [a15].
References
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Arguesian lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arguesian_lattice&oldid=38973