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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 | ||

+ | )] | ||

+ | |||

+ | |||

+ | |||

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