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 non-empty 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:

• for any $J\subseteq I$ and $x\in X^I$ , define the subset (of $X^I$ ) $$[x]_J:=\lbrace y\in X^I \mid x_i=y_i\mbox{ for every }i\notin J\rbrace,$$
• 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 $\tau:I\to I$ .

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 $(A,X,I)$ .

• $(B,I,\exists,S)$ is a polyadic algebra. The proof of this is not difficult, but involved, and can be found in the reference below.
• 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 $X$ .
• If $I$ is $\varnothing$ , then $(B,I,\exists,S)$ can be identified with the Boolean algebra $A$ , for $S=\varnothing$ and $X^I$ is a singleton, and hence the set of functions from $X^I$ to $A$ is identified with $A$ .

Bibliography

1

P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).