Semi-modular lattice

2020 Mathematics Subject Classification: Primary: 06B [MSN][ZBL]

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, cf Rank of a partially ordered set. A semi-modular lattice of finite length is a relatively complemented lattice if and only if it is 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 [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. For lattices of finite length, modularity is equivalent to upper and lower semi-modularity, but for lattices of infinite length this need not hold.

References

[1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)
[2]	F. Maeda, S. Maeda, "Theory of symmetric lattices" , Springer (1970)
[a1]	Manfred Stern, "Semimodular Lattices: Theory and Applications", Encyclopedia of Mathematics and its Applications 73, Cambridge University Press (1999) ISBN 0-521-46105-7