# Differences

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cylindric_algebras [2010/07/29 15:46] (current)
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+=====Cylindric algebras=====
+
+Abbreviation: **CA$_\alpha$**
+
+====Definition====
+A \emph{cylindric algebra} of dimension $\alpha$ is a [[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.
+
+==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:
+$h(x ... y)=h(x) ... h(y)$
+
+====Definition====
+An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +...\rangle$ such that
+
+$...$ is ...:  $axiom$
+
+$...$ is ...:  $axiom$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+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.
+
+^[[Classtype]]                        |variety  |
+^[[Equational theory]]                |undecidable for $\alpha\ge 3$, decidable otherwise |
+^[[Quasiequational theory]]           | |
+^[[First-order theory]]               | |
+^[[Locally finite]]                   |no |
+^[[Residual size]]                    |unbounded |
+^[[Congruence distributive]]          |yes |
+^[[Congruence modular]]               |yes |
+^[[Congruence $n$-permutable]]        |yes, $n=2$ |
+^[[Congruence regular]]               |yes |
+^[[Congruence uniform]]               |yes |
+^[[Congruence extension property]]    |yes |
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]      | |
+^[[Amalgamation property]]            | |
+^[[Strong amalgamation property]]     | |
+^[[Epimorphisms are surjective]]      | |
+
+====Finite members====
+
+$\begin{array}{lr} + f(1)= &1\\ + f(2)= &\\ + f(3)= &\\ + f(4)= &\\ + f(5)= &\\ +\end{array}$
+$\begin{array}{lr} + f(6)= &\\ + f(7)= &\\ + f(8)= &\\ + f(9)= &\\ + f(10)= &\\ +\end{array}$
+
+
+====Subclasses====
+  [[Representable cylindric algebras]] subvariety
+
+
+====Superclasses====
+  [[Diagonal free cylindric algebras]] subreduct
+
+  [[Two-dimensional cylindric algebras]] subreduct
+
+
+====References====
+