Mathematical Structures: Cylindric algebras

# Cylindric algebras

http://mathcs.chapman.edu/structuresold/files/Cylindric_algebras.pdf
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\begin{document}
\textbf{\Large Cylindric algebras}

\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.

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}

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