Recall that given a triple where is a Boolean algebra, and are sets. we can construct a polyadic algebra called the functional polyadic algebra for . In this entry, we will construct an example of a polyadic algebra with equality called the functional polyadic algebra with equality from .

We start with a simpler structure. Let be an arbitrary Boolean algebra, and are sets. Let , the set of all -indexed -valued sequences, and , the set of all functions from to . Call the function the functional equality associated with , if for each , is the function defined by

Now, will have the additional structure of being a polyadic algebra. Start with a Boolean algebra , and let and be defined as in the last paragraph. Then, as stated above in the first paragraph, and illustrated in here, is a polyadic algebra (called the functional polyadic algebra for ). Using the just constructed, the quadruple is a functional equality algebra, and is called the functional polyadic algebra with equality for .

It is not hard to show that is an equality predicate on , and as a result is a polyadic algebra with equality.

Bibliography

1

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