cylindric algebra
A cylindric algebra is a quadruple $(B,V,\exists ,d)$, where $B$ is a Boolean algebra^{}, $V$ is a set whose elements we call variables, $\exists $ and $d$ are functions
$$\exists :V\to {B}^{B}\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}d:V\times V\to B$$ 
such that

1.
$(B,\exists x)$ is a monadic algebra for each $x\in V$,

2.
$\exists x\circ \exists y=\exists y\circ \exists x$ for any $x,y\in V$,

3.
${d}_{xx}=1$ for all $x\in V$,

4.
for any $x,y\in V$ with $x\ne y$, and any $a\in B$, we have the equality
$$\exists x(a\wedge {d}_{xy})\wedge \exists x\left({a}^{\prime}\wedge {d}_{xy}\right)=0$$ 
5.
for any $x,y,z\in V$ with $x\ne y$ and $x\ne z$, we have the equality
$$\exists x({d}_{xy}\wedge {d}_{xz})={d}_{yz}.$$
where $\exists x$ and ${d}_{xy}$ are the abbreviations for $\exists (x)$ and $d(x,y)$ respectively.
Basically, the first two conditions say that the $(B,V,\exists )$ portion of the cylindric algebra is very similar^{} to a quantifier algebra, except the domain is no longer the subsets of $V$, but the elements of $V$ instead. The function $d$ is the algebraic abstraction of equality. Condition 3 says that $x=x$ is always true, condition 4 says that the proposition^{} $a$ and its complement^{} ${a}^{\prime}$, where any occurrences of the variable $x$ are replaced by the variable $y$, distinct from $x$, is always false, while condition 5 says $y=z$ iff there is an $x$ such that $x=y$ and $x=z$.
Below are some elementary properties of $d$:

•
(symmetric^{} property) ${d}_{xy}={d}_{yx}$

•
(transitive property) ${d}_{xy}\wedge {d}_{yz}\le {d}_{xz}$

•
$\exists x({d}_{xy})=1$

•
$\exists x({d}_{yz})={d}_{yz}$ provided that $x\notin \{y,z\}$

•
if $x\ne y$, then

(a)
$\exists x({d}_{xy}\wedge {a}^{\prime})={(\exists x({d}_{xy}\wedge a))}^{\prime}$,

(b)
${d}_{xy}\wedge a={d}_{xy}\wedge \exists x(a\wedge {d}_{xy})$.

(a)
Remarks

1.
The dimension of a cylindric algebra $(B,V,\exists ,d)$ is the cardinality of $V$.

2.
From the definition above, a cylindric algebra is a twosorted structure^{}, with $B$ and $V$ as the two distinct universes^{}. However, it is often useful to view a cylindric algebra as a onesorted structure. The way to do this is to dispense with $V$ and identify each $\exists x$ as a unary operator on $B$, and each ${d}_{xy}$ as a constant in $B$. As a result, the cylindric algebra $(B,V,\exists ,d)$ becomes a Boolean algebra with possibly infinitely many operators:
$${(B,\exists x,{d}_{xy})}_{x,y\in V}.$$ 
3.
Let $L$ be a the language^{} of a first order logic, and $S$ a set of sentences^{} in $L$. Define $\equiv $ on $L$ so that
$$\phi \equiv \psi \mathit{\hspace{1em}}\text{iff}\mathit{\hspace{1em}}S\u22a2(\phi \leftrightarrow \psi ).$$ Then $\equiv $ is an equivalence relation^{} on $L$. For each formula^{} $\phi \in L$, let $[\phi ]$ be the equivalence class^{} containing $\phi $. Let $V$ be a countably infinite^{} set of variables available to $L$. Now, define operations^{} $\vee ,\wedge {,}^{\prime},\exists x,{d}_{xy}$ as follows:
$[\phi ]\vee [\psi ]$ $:=$ $[\phi \vee \psi ],$ (1) $\left[\phi \right]\wedge [\psi ]$ $:=$ $[\phi \wedge \psi ],$ (2) ${\left[\phi \right]}^{\prime}$ $:=$ $[\mathrm{\neg}\phi ],$ (3) $0$ $:=$ $[\mathrm{\neg}x=x],$ (4) $1$ $:=$ $[x=x],$ (5) $\exists x[\phi ]$ $:=$ $[\exists x\phi ],$ (6) ${d}_{xy}$ $:=$ $[x=y].$ (7) Then it can be shown that $(L/\equiv ,V,\exists ,d)$ is a cylindric algebra. Thus a cylindric algebra can be thought of as an “algebraization” of first order logic (with equality), much the same way as a Boolean algebra (LindenbaumTarski algebra) as the algebraic counterpart of propositional logic.
References
 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
 2 L. Henkin, J. D. Monk, A. Tarski, Cylindric Algebras, Part I., NorthHolland, Amsterdam (1971).
 3 J. D. Monk, Mathematical Logic, Springer, New York (1976).
 4 B. Plotkin, Universal Algebra^{}, Algebraic Logic, and Databases, Kluwer Academic Publishers (1994).
Title  cylindric algebra 

Canonical name  CylindricAlgebra 
Date of creation  20130322 17:51:21 
Last modified on  20130322 17:51:21 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03G15 
Classification  msc 06E25 
Related topic  PolyadicAlgebra 
Related topic  PolyadicAlgebraWithEquality 