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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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