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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.
 +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.
 +
 +==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====
 +
 +[(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
 +see also http://www.math.iastate.edu/maddux/papers/raca.ps
 +)]
 +
 +
 +