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semimodular lattice (Definition)

A lattice $ L$ is semimodular 1if for any $ a$ and $ b\in L$,

$\displaystyle a \wedge b \prec a$   implies$\displaystyle \quad b \prec a \vee b, $
where $ \prec$ denotes the covering relation in $ L$. Dually, a lattice $ L$ is said to be lower semimodular if for any $ a$ and $ b\in L$,
$\displaystyle b \prec a \vee b$   implies$\displaystyle \quad a \wedge b \prec a. $
A chain finite lattice is modular if and only if it is both semimodular and lower semimodular.

The smallest lattice which is semimodular but not modular is

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & 1 \ar@{-}[ld] \ar@{-}[d] \ar@{... ...}[rd] & c \ar@{-}[d] \ d \ar@{-}[rd] & & e \ar@{-}[ld] \ & 0 & } } \end{xy}$
since $ d \le a$ but $ a \wedge (c \vee d) \neq (a \wedge c) \vee d$.



Footnotes

...semimodular 1
Or upper semimodular, if one wants to stress the distinction with lower semimodular lattices.


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"semimodular lattice" is owned by mps. [ full author list (2) | owner history (1) ]
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See Also: modular lattice, incidence geometry

Other names:  upper semimodular lattice, lower semimodular lattice
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Cross-references: modular, chain finite, relation, covering, lattice
There are 3 references to this entry.

This is version 6 of semimodular lattice, born on 2005-08-02, modified 2007-04-12.
Object id is 7286, canonical name is SemimodularLattice.
Accessed 3226 times total.

Classification:
AMS MSC06C10 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Semimodular lattices, geometric lattices)

Pending Errata and Addenda
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Discussion
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where is notation defined? by zabrocki on 2005-08-19 11:19:47
I can't seem to find a definition of the notation <:
Does this mean 'covers'
Where is this?
-Mike
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