Cylindric algebras

http://mathcs.chapman.edu/structuresold/files/Cylindric_algebras.pdf

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\begin{document}
\textbf{\Large Cylindric algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Cylindric_algebras}{edit}

\abbreviation{CA$_\alpha$}

\begin{definition}
A \emph{cylindric algebra} of dimension $\alpha$ is a \href{Boolean_algebras_with_operators.pdf}{Boolean algebra with operators} $\mathbf{A}=\langle A,
\vee, 0, \wedge, 1, -, c_i, d_{ij}: i,j<\alpha\rangle$ such that for all $i,j<\alpha$

the $c_i$ are increasing: $x\le c_i x$

the $c_i$ semi-distribute over $\wedge$: $c_i(x\wedge c_i y) = c_i x\wedge c_i y$

the $c_i$ commute: $c_ic_j x=c_jc_i x$

the diagonals $d_{ii}$ equal the top element:  $d_{ii}=1$

$d_{ij}=c_k(d_{ik}\wedge d_{kj})$ for $k\ne i,j$

$c_i(d_{ij}\wedge x)\wedge c_i(d_{ij}\wedge -x)=0$ for $i\ne j$

Remark: This is a template.
Click on the 'Edit text of this page' link at the bottom to add some information to this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: 
$h(x ... y)=h(x) ... h(y)$
\end{morphisms}

\begin{definition}
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
  Classtype                       & variety \\\hline
  Equational theory               & undecidable for $\alpha\ge 3$, decidable otherwise\\\hline
  Quasiequational theory          & \\\hline
  First-order theory              & \\\hline
  Locally finite                  & no\\\hline
  Residual size                   & unbounded\\\hline
  Congruence distributive         & yes\\\hline
  Congruence modular              & yes\\\hline
  Congruence $n$-permutable       & yes, $n=2$\\\hline
  Congruence regular              & yes\\\hline
  Congruence uniform              & yes\\\hline
  Congruence extension property   & yes\\\hline
  Definable principal congruences & \\\hline
  Equationally def. pr. cong.     & \\\hline
  Amalgamation property           & \\\hline
  Strong amalgamation property    & \\\hline
  Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &\\
  f(3)= &\\
  f(4)= &\\
  f(5)= &\\
\end{array}$\qquad
$\begin{array}{lr}
  f(6)= &\\
  f(7)= &\\
  f(8)= &\\
  f(9)= &\\
  f(10)= &\\
\end{array}$

\end{finite_members}

\begin{subclasses}\ 

  \href{Representable_cylindric_algebras.pdf}{Representable cylindric algebras} subvariety

\end{subclasses}

\begin{superclasses}\ 

  \href{Diagonal_free_cylindric_algebras.pdf}{Diagonal free cylindric algebras} subreduct

  \href{Two-dimensional_cylindric_algebras.pdf}{Two-dimensional cylindric algebras} subreduct

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Maddux1991}
Roger Maddux, \emph{Introductory course on relation algebras, finite-dimensional cylindric algebras, and their interconnections}, Algebraic Logic (Proc. Conf. Budapest 1988) ed. by H. Andreka, J. D. Monk, and I. Nemeti, Colloq. Math. Soc. J. Bolyai 54 North-Holland Amsterdam, 1991, 361--392 
\url{http://www.math.iastate.edu/maddux/papers/raca.ps}

\end{thebibliography}


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