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Relation Theory Poset Lattice Modular Lattice Chain Finite

1	Poset
1	Relation Theory
2	Lattice
4	Modular Lattice
10	Chain Finite

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Semimodular Lattice

A lattice is semimodular ¹if for any 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

. Dually, a lattice

is said to be lower semimodular if for any

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.