Modular lattice

Dedekind lattice

A lattice in which the modular law is valid, i.e. if $ a \leq c $, then $ ( a + b ) c = a + bc $ for any $ b $. This requirement amounts to saying that the identity $ ( ac + b ) c = ac + bc $ is valid. Examples of modular lattices include the lattices of subspaces of a linear space, of normal subgroups (but not all subgroups) of a group, of ideals in a ring, etc. A lattice with a composition sequence is a modular lattice if and only if there exists on it a dimension function $ d $, i.e. an integer-valued function such that $ d ( x + y ) + d ( xy ) = d ( x ) + d ( y ) $ and such that if the interval $ [ a , b] $ is prime, it follows that $ d ( b ) = d ( a ) + 1 $. If $ w = a _ {1} ^ {(} 1) \dots a _ {m _ {1} } ^ {(} 1) = a _ {1} ^ {(} 2) \dots a _ {m _ {2} } ^ {(} 2) $, if none of the elements $ a _ {i} ^ {(} k) $ can be represented as a product of elements other than itself and if

$$ a _ {1} ^ {(} k) \dots a _ {i-} 1 ^ {(} k) a _ {i+} 1 ^ {(} k) \dots a _ {mk} ^ {(} k) \Nle a _ {i} ^ {(} k) , $$

then $ m _ {1} = m _ {2} $ and for any $ a _ {i} ^ {(} 1) $ it is possible to find an element $ a _ {j} ^ {(} 2) $ such that

$$ w = a _ {1} ^ {(} 1) \dots a _ {i-} 1 ^ {(} 1) a _ {j} ^ {(} 2) a _ {i+} 1 ^ {(} 1) \dots a _ {m _ {1} } ^ {(} 1) , $$

[3], [6]. Non-zero elements $ a _ {1} \dots a _ {n} $ of a modular lattice with a zero 0 are said to be independent if $ ( a _ {1} + \dots + a _ {i-} 1 + a _ {j+} 1 + \dots + a _ {n} ) a _ {i} = 0 $ for all $ i $. This definition makes it possible to generalize many properties of systems of linearly independent vectors [3], [5], [6]. If $ a _ {1} \dots a _ {n} $ are independent, their sum is denoted by $ a _ {1} \oplus \dots \oplus a _ {n} $. Ore's theorem: If a modular lattice has a composition sequence and if

$$ 1 = a _ {1} ^ {(} 1) \oplus \dots \oplus a _ {m _ {1} } ^ {(} 1) = \ a _ {1} ^ {(} 2) \oplus \dots \oplus a _ {m _ {2} } ^ {(} 2) , $$

none of the elements $ a _ {i} ^ {(} k) $ being representable as a sum of two independent elements, then $ m _ {1} = m _ {2} $ and for each $ a _ {i} ^ {(} 1) $ it is possible to find an element $ a _ {j} ^ {(} 2) $ such that

$$ 1 = a _ {1} ^ {(} 1) \oplus \dots \oplus a _ {i-} 1 ^ {(} 1) \oplus a _ {j} ^ {(} 2) \oplus a _ {i+} 1 ^ {(} 1) \oplus \dots \oplus a _ {m _ {1} } ^ {(} 1) , $$

[3], [6]. In the case of completely modular lattices (cf. also Complete Dedekind lattice), which must satisfy certain additional requirements, the theorems on independent elements and direct decompositions may be applied to infinite sets as well [4], [5]. Complemented modular lattices have been studied; these are modular lattices with a 0 and a 1 in which for each element $ x $ there exists at least one element $ y $( said to be a complement of the element $ x $) such that $ x + y = 1 $, $ xy = 0 $. A complemented modular lattice which has a composition sequence, is isomorphic to the modular lattice of all subspaces of a finite-dimensional linear space over some skew-field. A complemented completely modular lattice $ L $ is isomorphic to the modular lattice of all subspaces of a linear (not necessarily finite-dimensional) space over some skew-field if and only if the following conditions are met: a) if $ 0 \neq a \in L $, it is possible to find an atom $ p \leq a $; b) if $ p $ is an atom and $ p \leq \sup A $, where $ A \subseteq L $, then $ p \leq \sup F $ for some finite set $ F \subseteq A $; c) if $ p , q $ are distinct atoms, it is possible to find a third atom $ r \leq p+ q $; and d) there exist at least three independent atoms. The last condition d) may be replaced by the requirement that the Desargues assumption be valid [2]. A further generalization of this result, which leads to regular rings [7], [5], is connected with the theory of von Neumann algebras. For a modular lattice with a composition sequence the presence of complements is equivalent to the representability of the unit as a sum of atoms.

Modular lattices are (in the Soviet Union) also called Dedekind lattices, in honour of R. Dedekind, who was the first to formulate the modular law and established a number of its consequences [1].

References

[1]	R. Dedekind, "Ueber die von drei Moduln erzeugte Dualgruppe" Math. Ann. , 53 (1900) pp. 371–403
[2]	R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952)
[3]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[4]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[5]	L.A. Skornyakov, "Complemented modular lattices and regular rings" , Oliver & Boyd (1964) (Translated from Russian)
[6]	L.A. Skornyakov, "Elements of lattice theory" , Hindushtan Publ. Comp. (1977) (Translated from Russian)
[7]	J. von Neumann, "Continuous geometry" , Princeton Univ. Press (1960)

Comments

Modular lattices satisfying the Desargues assumption are called Desarguesian lattices. Complemented completely modular lattices satisfying the identity

$$ a \sum _ {i \in I } b _ {i} = \ \sum _ {i \in I } a b _ {i} $$

whenever the set $ \{ {b _ {i} } : {i \in I } \} $ is (upwards) directed, and the dual of the latter condition, are called continuous geometries [5], [7].

The "dimension" is also called rank, cf Rank of a partially ordered set; a "prime" interval is an elementary interval.