Semi-modular association

The hexagon lattice S ₇ with a center , also known as D ₂ , is semimodular but not modular.

This article deals with generalizations of modularity, which are defined using the coverage relation. For M-symmetry, the generalization of modularity using modular pairs, see modular lattice .

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

Semi-modular law

${\displaystyle a\sqcap b\lessdot a}$implies . ${\displaystyle b\lessdot a\sqcup b}$

The notation means that the element covers the element , i.e. H. and for all elements with true or . ${\displaystyle a\lessdot b}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle a<b}$ ${\displaystyle c}$ ${\displaystyle a\leq c\leq b}$ ${\displaystyle c=a}$ ${\displaystyle c=b}$

An atomic (and therefore algebraic ) semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to (simple) matroids . An atomistic semimodular bounded lattice of finite length is called a geometrical lattice and corresponds to a matroid of finite rank. (These definitions follow Stern (1999). Some authors use the term "geometric lattice" for the more general matroid lattices. But most authors only consider the finite case in which both definitions of "semimodular and atomistic" are equivalent.)

A finite lattice is modular if and only if both it and the dual lattice are semimodular. (Semimodular associations are also called upper semimodular in English ; the dual term then means lower semimodular .)

A finite lattice, or more generally a lattice that fulfills the ascending chain condition or the descending chain condition, is semimodular if and only if it is M-symmetric . Some authors refer to M-symmetrical associations as semi-modular associations. (E.g. Fofanova (2001).)

The Dedekind chain theorem applies to every (up or down) semi-modular association .

Birkhoff's condition

A bandage is sometimes called weakly semimodular if it meets the following condition, which goes back to Garrett Birkhoff :

Birkhoff's condition

If and is, is and . ${\displaystyle a\sqcap b\lessdot a}$ ${\displaystyle a\sqcap b\lessdot b}$ ${\displaystyle a\lessdot a\sqcup b}$ ${\displaystyle b\lessdot a\sqcup b}$

Every semi-modular association is weakly semi-modular. The reverse is true for lattices of finite length, and more generally for relatively atomic lattices that are continuous above.

Mac Lane's condition

The following two conditions are equivalent for all associations. They were found by Saunders Mac Lane when he was looking for a condition that is equivalent to semimodularity for finite lattices but does not use the coverage relation.

Mac Lane's condition 1

For everyone with there is an element such that and . ${\displaystyle a,b,c}$ ${\displaystyle b\sqcap c<a<c<b\sqcup a}$ ${\displaystyle d\leq b}$ ${\displaystyle b\sqcap c<d}$ ${\displaystyle a=(a\sqcup d)\sqcap c}$

Mac Lane's condition 2

For everyone with there is an element such that and . ${\displaystyle a,b,c}$ ${\displaystyle b\sqcap c<a<c<b\sqcup c}$ ${\displaystyle d\leq b}$ ${\displaystyle b\sqcap c<d}$ ${\displaystyle a=(a\sqcup d)\sqcap c}$

Any association that meets Mac Lane's condition (s) is semi-modular. The converse is true for lattices of finite length, and more generally for relatively atomic lattices. In addition, any above continuous lattice that meets Mac Lane's conditions is M-symmetric.

literature

• Garrett Birkhoff : Lattice Theory (=  American Mathematical Society Colloquium Publications . Volume XXV ). 3. Edition. American Mathematical Society , Providence, RI 1967 ( MR0227053 ).
• Fofanova, TS (2001), "Semi-modular lattice" , in Hazewinkel, Michiel, Encyclopaedia of Mathematics , Kluwer Academic Publishers, ISBN 978-1556080104 . (The article covers M-symmetric bandages.)
• Helmuth Gericke : Theory of Associations (=  university pocket books. 38 / 38a). Bibliographisches Institut , Mannheim 1967 ( MR0219453 ).
• Stern, Manfred (1999), Semimodular lattices , Cambridge University Press, ISBN 978-0-521-46105-4 .

Individual evidence

1. ↑ Helmuth Gericke: Theory of Associations. 1967, p. 68 ff.