Arguesian lattice

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].

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