Differences

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]]
		+	)]