Differences
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lattices [2010/08/01 16:21] 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: $x\vee y=y\vee x$, $x\wedge y=y\wedge x$ | ||
| Line 26: | Line 26: | ||
| ==Morphisms== | ==Morphisms== | ||
| Let $\mathbf{L}$ and $\mathbf{M}$ be lattices. A morphism from $\mathbf{L}$ | Let $\mathbf{L}$ and $\mathbf{M}$ be lattices. A morphism from $\mathbf{L}$ | ||
| - | to $\mathbf{M}$ is a function $h:Larrow M$ that is a homomorphism: | + | to $\mathbf{M}$ is a function $h:L\to M$ that is a homomorphism: |
| $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$ | $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$ | ||
| Line 37: | Line 37: | ||
| $\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: $(x\vee y)\wedge x=x$, $(x\wedge y)\vee x=x$ | ||
| Line 45: | Line 45: | ||
| 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: $z=x\wedge y\Longleftrightarrow z\leq x$, $z\leq y\ \mbox{and}\ \forall w\ (w\leq x$, $w\leq y\Longrightarrow w\leq z)$ |
| Line 55: | Line 55: | ||
| 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: $x\leq z\Longrightarrow x\wedge y\leq z$, $\quad y\leq z\Longrightarrow x\wedge y\leq z$ | ||
| Line 74: | Line 74: | ||
| subsets of a sets $S$, ordered by inclusion. | subsets of a sets $S$, ordered by inclusion. | ||
| - | !-[[Lattices (mace)]]-! | + | /*[[Lattices (mace)]]*/ |
| Line 122: | Line 122: | ||
| 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==== | ||
| Line 155: | Line 156: | ||
| ====References==== | ====References==== | ||
| - | |||
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