Semi-modular lattice

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 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). 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 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 [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

[1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)
[2]	F. Maeda, S. Maeda, "Theory of symmetric lattices" , Springer (1970)