example of polyadic algebra
Recall that the canonical example of a monadic algebra is that of a functional monadic algebra, which is a pair $(B,\exists )$ such that $B$ is the set of all functions^{} from a nonempty set $X$ to a Boolean algebra^{} $A$ such that, for each $f\in B$, the supremum^{} and the infimum^{} of $f(X)$ exist, and $\exists $ is a function on $B$ that maps each element $f$ to ${f}^{\exists}$, a constant element whose range is a singleton consisting of the supremum of $f(X)$.
The canonical example of a polyadic algebra is an extension^{} (generalization^{}) of a functional monadic algebra, known as the functional polyadic algebra. Instead of looking at functions from $X$ to $A$, we look at functions from ${X}^{I}$ (where $I$ is some set), the $I$fold cartesian power of $X$, to $A$. In this entry, an element $x\in {X}^{I}$ is written as a sequence of elements of $A$: ${({x}_{i})}_{i\in I}$ where ${x}_{i}\in A$, or $({x}_{i})$ for short.
Before constructing the functional polyadic algebra based on the sets $X,I$ and the Boolean algebra $A$, we first introduce the following notations:

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for any $J\subseteq I$ and $x\in {X}^{I}$, define the subset (of ${X}^{I}$)
$${[x]}_{J}:=\{y\in {X}^{I}\mid {x}_{i}={y}_{i}\text{for every}i\notin J\},$$ 
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for any function $\tau :I\to I$ and any $f:{X}^{I}\to A$, define the function ${f}_{\tau}$ from ${X}^{I}$ to $A$, given by
$${f}_{\tau}({x}_{i}):=f({x}_{\tau (i)}).$$
Now, let $B$ be the set of all functions from ${X}^{I}$ to $A$ such that

1.
for every $f\in B$, every $J\subseteq I$ and every $x\in {X}^{I}$, the arbitrary join
$$\bigvee f\left({[x]}_{J}\right)$$ exists.
Before stating the next condition, we introduce, for each $f\in B$, a function ${f}^{\exists J}:{X}^{I}\to A$ as follows:
$${f}^{\exists J}(x):=\bigvee f\left({[x]}_{J}\right).$$ Now, we are ready for the next condition:

2.
if $f\in B$, then ${f}^{\exists J}\in B$,

3.
if $f\in B$, then ${f}_{\tau}\in B$ for $\tau :I\to I$.
Note that if $A$ were a complete Boolean algebra, we can take $B$ to be ${A}^{{X}^{I}}$, the set of all functions from ${X}^{I}$ to $A$.
Next, define $\exists :P(I)\to {B}^{B}$ by $\exists (J)(f)={f}^{\exists J}$, and let $S$ be the semigroup of functions on $I$ (with functional^{} compositions as multiplications^{}), then we call the quadruple $(B,I,\exists ,S)$ the functional polyadic algebra for the triple $(A,X,I)$.
Remarks. Let $(B,I,\exists ,S)$ be the functional polyadic algebra for $(A,X,I)$.

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$(B,I,\exists ,S)$ is a polyadic algebra. The proof of this is not difficult, but involved, and can be found in the reference below.

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If $I$ is a singleton, then $(B,I,\exists ,S)$ can be identified with the functional monadic algebra $(B,\exists )$ for $(A,X)$, for $S$ is just $I$, and ${X}^{I}$ is just $X$.

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If $I$ is $\mathrm{\varnothing}$, then $(B,I,\exists ,S)$ can be identified with the Boolean algebra $A$, for $S=\mathrm{\varnothing}$ and ${X}^{I}$ is a singleton, and hence the set of functions from ${X}^{I}$ to $A$ is identified with $A$.
References
 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
Title  example of polyadic algebra 

Canonical name  ExampleOfPolyadicAlgebra 
Date of creation  20130322 17:53:20 
Last modified on  20130322 17:53:20 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  15 
Author  CWoo (3771) 
Entry type  Example 
Classification  msc 03G15 
Defines  functional polyadic algebra 