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Homesemimodular lattice

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# semimodular lattice

A lattice $L$ is semimodular
^{1}^{1}Or upper semimodular, if one wants to stress the
distinction with lower semimodular lattices.
if for any $a$ and $b\in L$,

$a\wedge b\prec a\quad\text{implies}\quad b\prec a\vee b,$ |

where $\prec$ denotes the covering relation in $L$.
Dually, a lattice $L$ is said to be *lower semimodular*
if for any $a$ and $b\in L$,

$b\prec a\vee b\quad\text{implies}\quad a\wedge b\prec a.$ |

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

$\xymatrix{&1\ar@{-}[ld]\ar@{-}[d]\ar@{-}[rd]&\\ a\ar@{-}[d]&b\ar@{-}[ld]\ar@{-}[rd]&c\ar@{-}[d]\\ d\ar@{-}[rd]&&e\ar@{-}[ld]\\ &0&}$ |

since $d\leq a$ but $a\wedge(c\vee d)\neq(a\wedge c)\vee d$.

Related:

ModularLattice, IncidenceGeometry

Synonym:

upper semimodular lattice, lower semimodular lattice

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

06C10*no label found*

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## Corrections

just need some clarification by CWoo ✓

semimodular lattice by noio ✓

covering relation by CWoo ✓

in addition by ixionid ✓

semimodular lattice by noio ✓

covering relation by CWoo ✓

in addition by ixionid ✓

## Comments

## where is notation defined?

I can't seem to find a definition of the notation <:

Does this mean 'covers'

Where is this?

-Mike

## Re: where is notation defined?

Yes, it means "covers".

My apologies if this was unclear, but I mostly

wrote down a quick definition of semimodular

lattice so that I could refer to it in a correction

posted to "incidence geometry". This may well

have been an end left loose in it.

I have (off-line) a half-finished entry

which defines this (and much other notation

for relations), which I have intended to

complete for a couple of weeks now, but I have

gotten sidetracked by work.