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lattices [2010/08/01 16:23]
jipsen
lattices [2016/01/27 11:48] (current)
jipsen
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called the \emph{join} and \emph{meet}, such that called the \emph{join} and \emph{meet}, such that
-$\vee ,\wedge $ are associative:  $(x\vee y)\vee z=x\vee (y\vee z)$,$\ (x\wedge y)\wedge z=x\wedge (y\wedge z)$+$\vee ,\wedge $ are associative:  $(x\vee y)\vee z=x\vee (y\vee z)$, $(x\wedge y)\wedge z=x\wedge (y\wedge z)$
$\vee ,\wedge $ are commutative:  $x\vee y=y\vee x$, $x\wedge y=y\wedge x$ $\vee ,\wedge $ are commutative:  $x\vee y=y\vee x$, $x\wedge y=y\wedge x$
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$\langle L,\vee \rangle $ and $\langle L,\wedge $\langle L,\vee \rangle $ and $\langle L,\wedge
\rangle $ are \rangle $ are
-[[Semilattices]], and+[[semilattices]], and
$\vee ,\wedge $ are absorbtive:  $(x\vee y)\wedge x=x$, $(x\wedge y)\vee x=x$ $\vee ,\wedge $ are absorbtive:  $(x\vee y)\wedge x=x$, $(x\wedge y)\vee x=x$
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A \emph{lattice} is a structure $\mathbf{L}=\langle L,\leq A \emph{lattice} is a structure $\mathbf{L}=\langle L,\leq
\rangle $ that is a \rangle $ that is a
-[[Partially ordered sets]] in which all elements $x,y\in L$ have a+[[partially ordered set]] in which all elements $x,y\in L$ have a
-least upper bound:  $z=x\vee y\Longleftrightarrow x\leq z$, $y\leq z \mbox{and}\ \forall w\ (x\leq w$, $y\leq w\Longrightarrow z\leq w)$+least upper bound:  $z=x\vee y\Longleftrightarrow x\leq z$, $y\leq z\ \mbox{and}\ \forall w\ (x\leq w$, $y\leq w\Longrightarrow z\leq w)$ and a
-greatest lower bound:  $z=x\wedge y\Longleftrightarrow z\leq x$, $z\leq y \mbox{and}\ \forall w\ (w\leq x$, $w\leq y\Longrightarrow w\leq z)$+greatest lower bound:  $z=x\wedge y\Longleftrightarrow z\leq x$, $z\leq y\ \mbox{and}\ \forall w\ (w\leq x$, $w\leq y\Longrightarrow w\leq z)$
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A \emph{lattice} is a structure $\mathbf{L}=\langle L,\vee ,\wedge A \emph{lattice} is a structure $\mathbf{L}=\langle L,\vee ,\wedge
,\leq \rangle $ such that $\langle L,\leq \rangle $ is a ,\leq \rangle $ such that $\langle L,\leq \rangle $ is a
-[[Partially ordered sets]] and the following quasiequations hold:+[[partially ordered set]] and the following quasiequations hold:
-$\vee $-left:  $x\leq z$, $y\leq z\ \Longrightarrow x\vee y\leq z$+$\vee $-left:  $x\leq z$ and $y\leq z\ \Longrightarrow x\vee y\leq z$
$\vee $-right:  $z\leq x\Longrightarrow z\leq x\vee y$, $\quad z\leq y\Longrightarrow z\leq x\vee y$ $\vee $-right:  $z\leq x\Longrightarrow z\leq x\vee y$, $\quad z\leq y\Longrightarrow z\leq x\vee y$
-$\wedge $-right:  $z\leq x$, $z\leq y\Longrightarrow z\leq x\wedge y$+$\wedge $-right:  $z\leq x$ and $z\leq y\Longrightarrow z\leq x\wedge y$
$\wedge $-left:  $x\leq z\Longrightarrow x\wedge y\leq z$, $\quad y\leq z\Longrightarrow x\wedge y\leq z$ $\wedge $-left:  $x\leq z\Longrightarrow x\wedge y\leq z$, $\quad y\leq z\Longrightarrow x\wedge y\leq z$
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subsets of a sets $S$, ordered by inclusion. subsets of a sets $S$, ordered by inclusion.
-[[Lattices (mace)]]-!+/*[[Lattices (mace)]]*/
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f(17)= &15150569446\\ f(17)= &15150569446\\
f(18)= &165269824761\\ f(18)= &165269824761\\
-\end{array}$ +f(19)= &1901910625578 
- +\end{array}$[(Jobst Heitzig, J\"urgen Reinhold, \emph{Counting finite lattices},
-[(Jobst Heitzig, J\"urgen Reinhold, \emph{Counting finite lattices},+
Algebra Universalis, Algebra Universalis,
-\textbf{48}, 2002, 43--53)]+\textbf{48}, 2002, 43--53)][(Peter Jipsen, Nathan Lawless, \emph{Generating all finite modular lattices of a given size},  
 +Algebra Universalis, \textbf{74}, 2015, 253--264)]
-!-[[Lattices of size 1 to 6]] 
-[[Finite lattices]] of size $\le 7$+[[http://math.chapman.edu/~jipsen/posets/lattices77.html|Diagrams of lattices of size 2 to 7]] 
 + 
 +/*[[Finite lattices]] of size $\le 7$
[[Subdirectly irreducible lattices]] of size $\le 7$ [[Subdirectly irreducible lattices]] of size $\le 7$
[[Lattices not in some subclasses]] of size $\le 7$ [[Lattices not in some subclasses]] of size $\le 7$
--!+*/
====Subclasses==== ====Subclasses====
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====References==== ====References====
-