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polyadic algebra
A polyadic algebra is a quadruple $(B,V,\exists,S)$, where $(B,V,\exists)$ is a quantifier algebra, and $S$ is a function from the set of functions on $V$ to the set of endomorphisms on the Boolean algebra $B$, in other words
$S:V^{V}\to\operatorname{End}(B)$ 
such that
1. $S(1_{V})=1_{B}$,
2. $S(f\circ g)=S(f)\circ S(g)$,
3. $S(f)\circ\exists(I)=S(g)\circ\exists(I)$ if $f(VI)=g(VI)$,
4. $\exists(I)\circ S(f)=S(f)\circ\exists(f^{{1}}(I))$ if $f$ is onetoone when restricted to $f^{{1}}(I)$.
Explanation of notations: $1_{V},1_{B}$ are identity functions on $V,B$ respectively; $f,g$ are functions on $V$, and $I$ is a subset of $V$. The circle $\circ$ represents functional compositions.
The degree and local finiteness of a polyadic algebra are defined as the degree and local finiteness of the underlying quantifier algebra.
Heuristically, the function $S$ can be thought of as changes to propositional functions due to a “substitution” of variables (elements of $V$). Let us see some examples. Let $V=\{x_{0},x_{1},\ldots\}$ be a countably indexed set of variables. For any propositional function $\varphi$, define $S(f)(\varphi)$ to be the propositional function $\varphi_{1}$ obtained by replacing each variable $x$ that occurs in it by $f(x)$. Below are two examples illustrating how $S(f)$ changes propositional functions:

Let $f:V\to V$ be the function given by $f(x_{0})=f(x_{1})=x_{0}$ and $f(x_{i})=x_{{i+1}}$ for all $i>1$. If $\varphi$ is the propositional function $x_{0}^{2}x_{1}+x_{2}/x_{3}$, then $S(f)(\varphi)$ is the propositional function $x_{0}^{2}x_{0}+x_{3}/x_{4}$.

Let $f:V\to V$ be the function given by $f(x_{0})=x_{2}$, and $f(x_{i})=x_{i}$ for all $i\neq 0$. Then the propositional function “$\exists x_{0},x_{1},x_{2}(x_{0}\neq x_{1}\wedge x_{1}\neq x_{2}\wedge x_{2}% \neq x_{0})$” becomes “$\exists x_{2},x_{1},x_{2}(x_{2}\neq x_{1}\wedge x_{1}\neq x_{2}\wedge x_{2}% \neq x_{2})$” under $S(f)$.
It is not hard to see from the examples above that $S(f)$ respects Boolean operations $\wedge$ and ${}^{{\prime}}$, which is why we want to make $S(f)$ an endomorphism on $B$. Furthermore, the four conditions above can be interpreted as
1. if there are no substitutions of variables, then there should be no corresponding changes to the propositional functions
2. applying substitutions $f\circ g$ of varaibles in a propositional function $\varphi$ should have the same effect as applying substitutions $g$ of variables in $\varphi$, followed by substitutions $f$ of variables in $S(g)(\varphi)$
3. a substitution $f$ of variables should have no effect to a propositional function beginning with $\exists$ if every variable bound by $\exists$ is fixed by $f$. For example, if we replace $f$ in the second example above by $f(x_{3})=x_{2}$ and $f(x_{i})=x_{i}$ otherwise, then
$``\exists x_{0},x_{1},x_{2}(x_{0}\neq x_{1}\wedge x_{1}\neq x_{2}\wedge x_{2}% \neq x_{0})"$ is unchanged by $S(f)$, since $x_{0},x_{1},x_{2}$ are all fixed by $f$.
4. Let $\varphi=\exists I\psi(I,J)$ be a propositional function, where $I,J$ are sets of variables with $I$ bound by $\exists$ and $J$ free. If no two variables $I$ get mapped to the same variable, and no free variable (in $J$) becomes bound (in $f(I)$) under the substitution, then $\exists I\psi(f(I),f(J))$ and $\exists f(I)\psi(f(I),f(J))$ are semantically the same, which is exactly the statement in the condition.
Remarks.

Paul Halmos first introduced the notion of polyadic algebras. In his Algebraic Logic, a compilation of articles on the subject, he called a function on the set $V$ of variables a transformation, and the triple $(B,V,S)$ satisfying the first two conditions above a transformation algebra. Therefore, a polyadic algebra is a quadruple $(B,V,\exists,S)$ where $(B,V,\exists)$ is a quantifier algebra and $(B,V,S)$ is a transformation algebra, such that conditions 3 and 4 above are satisfied.

The notion of polyadic algebras generalizes the notion of monadic algebras. Indeed, a monadic algebra is a polyadic algebra where $V$ is a singleton.

Just as a LindenbaumTarski algebra is the algebraic counterpart of a classical propositional logic, a polyadic algebra is the algebraic counterpart of a classical first order logic without equality. A variant of the polyadic algebra is what is known as a cylindric algebra, which algebratizes a classical first order logic with equality.
References
 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
 2 B. Plotkin, Universal Algebra, Algebraic Logic, and Databases, Kluwer Academic Publishers (1994).
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