View source for Modular lattice

''Dedekind lattice''

A [[Lattice|lattice]] in which the modular law is valid, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m0644601.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m0644602.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m0644603.png" />. This requirement amounts to saying that the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m0644604.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m0644605.png" />, i.e. an integer-valued function such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m0644606.png" /> and such that if the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m0644607.png" /> is prime, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m0644608.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m0644609.png" />, if none of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446010.png" /> can be represented as a product of elements other than itself and if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446011.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446012.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446013.png" /> it is possible to find an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446014.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446015.png" /></td> </tr></table>

[[#References|[3]]], [[#References|[6]]]. Non-zero elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446016.png" /> of a modular lattice with a zero 0 are said to be independent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446018.png" />. This definition makes it possible to generalize many properties of systems of linearly independent vectors [[#References|[3]]], [[#References|[5]]], [[#References|[6]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446019.png" /> are independent, their sum is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446020.png" />. Ore's theorem: If a modular lattice has a composition sequence and if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446021.png" /></td> </tr></table>

none of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446022.png" /> being representable as a sum of two independent elements, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446023.png" /> and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446024.png" /> it is possible to find an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446025.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446026.png" /></td> </tr></table>

[[#References|[3]]], [[#References|[6]]]. In the case of completely modular lattices (cf. also [[Complete Dedekind lattice|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 [[#References|[4]]], [[#References|[5]]]. Complemented modular lattices have been studied; these are modular lattices with a 0 and a 1 in which for each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446027.png" /> there exists at least one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446028.png" /> (said to be a complement of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446029.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446031.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446032.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446033.png" />, it is possible to find an [[Atom|atom]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446034.png" />; b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446035.png" /> is an atom and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446038.png" /> for some finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446039.png" />; c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446040.png" /> are distinct atoms, it is possible to find a third atom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446041.png" />; and d) there exist at least three independent atoms. The last condition d) may be replaced by the requirement that the [[Desargues assumption|Desargues assumption]] be valid [[#References|[2]]]. A further generalization of this result, which leads to regular rings [[#References|[7]]], [[#References|[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 [[#References|[1]]].

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



====Comments====
Modular lattices satisfying the [[Desargues assumption]] are called Desarguesian lattices. Complemented completely modular lattices satisfying the identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446042.png" /></td> </tr></table>

whenever the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064460/m06446043.png" /> is (upwards) directed, and the dual of the latter condition, are called continuous geometries [[#References|[5]]], [[#References|[7]]].

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