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lattices [2010/08/01 16:34] jipsen |
lattices [2016/01/27 11:48] (current) jipsen |
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| [[partially ordered set]] 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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| 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 $\le 7$ | ||
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| ====References==== | ====References==== | ||
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