Semimodular lattice

In order theory is meant by a semi- modular lattice an association that satisfies the following condition:

The notation indicates that the element covering the element, ie and applies to all elements with or.

An atomic (and hence algebraic ) semi modular limited association called Matroidverband because such associations to (simple) matroids are equivalent. An atomistic semi modular limited association of finite length is called geometric association and corresponds to a matroid of finite rank. ( These definitions follow Stern (1999). Some authors use the term " geometric association " for the general Matroidverbände. , But most authors consider only the finite case, in which both definitions of "semi modular and atomistic " are equivalent. )

A finite lattice is modular then when both he and the dual association is semi- modular. (Semi Modular associations are known in English as upper semi- modular, the dual concept is then called lower semi- modular. )

A finite union, or more generally an association satisfies the ascending chain condition or the descending chain condition is only semi- modular, if it is M- symmetric. Some authors refer to M- symmetric associations as semi-modular organizations. ( For example Fofanova ( 2001). )

Birkhoff condition

An association is sometimes weak semi modular mentioned if it meets the following goes back to Garrett Birkhoff condition:

Each semi-modular bandage is weakly semi- modular. The converse is true for organizations of finite length, and more generally for steady upward relative atomic associations.

Mac Lane's condition

The following two conditions are equivalent for all associations. They were found by Saunders Mac Lane, when he sought a condition, but that is not for finite associations to Semimodularität equivalent used the cover relation.

Each association that Mac Lane's condition is fulfilled (s) is semi- modular. The converse is true for organizations of finite length, and more generally for relatively atomic associations. In addition, each top continuous association of Mac Lane's conditions is fulfilled M- symmetric.