example of polyadic algebra with equality
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
The quadruple is called a functional equality algebra.
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 (http://planetmath.org/ExampleOfPolyadicAlgebra), 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.
- 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
|Title||example of polyadic algebra with equality|
|Date of creation||2013-03-22 17:55:24|
|Last modified on||2013-03-22 17:55:24|
|Last modified by||CWoo (3771)|
|Defines||functional equality algebra|
|Defines||functional polyadic algebra with equality|