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bounded_lattices [2010/07/29 15:19] jipsen created |
bounded_lattices [2010/09/04 16:55] (current) jipsen delete hyperbaseurl |
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- | f | + | =====Bounded lattices===== |
+ | |||
+ | Abbreviation: **BLat** | ||
+ | |||
+ | ====Definition==== | ||
+ | A \emph{bounded lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,1\rangle$ such that | ||
+ | |||
+ | $\langle L,\vee,\wedge\rangle $ is a [[lattice]] | ||
+ | |||
+ | $0$ is the least element: $0\leq x$ | ||
+ | |||
+ | $1$ is the greatest element: $x\leq 1$ | ||
+ | ==Morphisms== | ||
+ | Let $\mathbf{L}$ and $\mathbf{M}$ be bounded lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a | ||
+ | homomorphism: | ||
+ | |||
+ | $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$ | ||
+ | |||
+ | ====Examples==== | ||
+ | Example 1: | ||
+ | |||
+ | ====Basic results==== | ||
+ | |||
+ | |||
+ | ====Properties==== | ||
+ | ^[[Classtype]] |variety | | ||
+ | ^[[Equational theory]] |decidable | | ||
+ | ^[[Quasiequational theory]] |decidable | | ||
+ | ^[[First-order theory]] |undecidable | | ||
+ | ^[[Congruence distributive]] |yes | | ||
+ | ^[[Congruence modular]] |yes | | ||
+ | ^[[Congruence n-permutable]] |no | | ||
+ | ^[[Congruence regular]] |no | | ||
+ | ^[[Congruence uniform]] |no | | ||
+ | ^[[Congruence extension property]] |no | | ||
+ | ^[[Definable principal congruences]] |no | | ||
+ | ^[[Equationally def. pr. cong.]] |no | | ||
+ | ^[[Amalgamation property]] |yes | | ||
+ | ^[[Strong amalgamation property]] |yes | | ||
+ | ^[[Epimorphisms are surjective]] |yes | | ||
+ | ^[[Locally finite]] |no | | ||
+ | ^[[Residual size]] |unbounded | | ||
+ | |||
+ | ====Finite members==== | ||
+ | |||
+ | $\begin{array}{lr} | ||
+ | f(1)= &1\\ | ||
+ | f(2)= &1\\ | ||
+ | f(3)= &1\\ | ||
+ | f(4)= &2\\ | ||
+ | f(5)= &5\\ | ||
+ | \end{array}$ | ||
+ | $\begin{array}{lr} | ||
+ | f(6)= &15\\ | ||
+ | f(7)= &53\\ | ||
+ | f(8)= &222\\ | ||
+ | f(9)= &1078\\ | ||
+ | f(10)= &5994\\ | ||
+ | \end{array}$ | ||
+ | $\begin{array}{lr} | ||
+ | f(11)= &37622\\ | ||
+ | f(12)= &262776\\ | ||
+ | f(13)= &2018305\\ | ||
+ | f(14)= &16873364\\ | ||
+ | f(15)= &152233518\\ | ||
+ | \end{array}$ | ||
+ | $\begin{array}{lr} | ||
+ | f(16)= &1471613387\\ | ||
+ | f(17)= &15150569446\\ | ||
+ | f(18)= &165269824761\\ | ||
+ | f(19)= &\\ | ||
+ | f(20)= &\\ | ||
+ | \end{array}$ | ||
+ | |||
+ | [(HeiRei2002)] | ||
+ | |||
+ | |||
+ | ====Subclasses==== | ||
+ | [[Bounded modular lattices]] | ||
+ | |||
+ | [[Complete lattices]] | ||
+ | |||
+ | |||
+ | ====Superclasses==== | ||
+ | [[Lattices]] | ||
+ | |||
+ | |||
+ | ====References==== | ||
+ | |||
+ | [(HeiRei2002> | ||
+ | Jobst Heitzig and J\"urgen Reinhold, \emph{Counting finite lattices}, | ||
+ | Algebra Universalis, | ||
+ | \textbf{48}, 2002, 43--53 [[MRreview]] | ||
+ | )] |
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