View source for Semi-modular lattice

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''semi-Dedekind lattice''

A lattice in which the modularity relation is symmetric, i.e. $aMb$ implies $bMa$ for any lattice elements $a,b$. The modularity relation here is defined as follows: Two elements $a$ and $b$ are said to constitute a modular pair, in symbols $aMb$, if $a(b+c)=ab+c$ for any $c\leq a$. A lattice in which every pair of elements is modular is called a [[Modular lattice|modular lattice]] or a Dedekind lattice.

A lattice of finite length is a semi-modular lattice if and only if it satisfies the covering condition: If $x$ and $y$ cover $xy$, then $x+y$ covers $x$ and $y$ (see [[Covering element|Covering element]]). In any semi-modular lattice of finite length one has the Jordan–Dedekind chain condition (all maximal chains between two fixed elements are of the same length; this makes it possible to develop a theory of dimension in such lattices. A semi-modular lattice of finite length is a relatively complemented lattice if and only if it a [[Atomistic lattice|atomistic]]: each of its elements is a union of atoms. Such lattices are known as ''geometric lattices''. An important class of semi-modular lattices is that of the  "nearly geometric"  matroid lattices (see [[#References|[2]]]). Every finite lattice is isomorphic to a sublattice of a finite semi-modular lattice. The class of semi-modular lattices is not closed under taking homomorphic images.

Besides semi-modular lattices, which are also known as upper semi-modular lattices, one also considers lower semi-modular lattices, which are defined in dual fashion. Examples of semi-modular lattices, apart from modular lattices, are the lattices of all partitions of finite sets and the lattices of linear varieties of affine spaces.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,   "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1967)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  F. Maeda,   S. Maeda,   "Theory of symmetric lattices" , Springer  (1970)</TD></TR>
</table>