bounded_lattices - MathStructures

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=====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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