example of polyadic algebra
Recall that the canonical example of a monadic algebra is that of a functional monadic algebra, which is a pair such that is the set of all functions from a non-empty set to a Boolean algebra such that, for each , the supremum and the infimum of exist, and is a function on that maps each element to , a constant element whose range is a singleton consisting of the supremum of .
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 to , we look at functions from (where is some set), the -fold cartesian power of , to . In this entry, an element is written as a sequence of elements of : where , or for short.
Before constructing the functional polyadic algebra based on the sets and the Boolean algebra , we first introduce the following notations:
for any and , define the subset (of )
for any function and any , define the function from to , given by
Now, let be the set of all functions from to such that
for every , every and every , the arbitrary join
Before stating the next condition, we introduce, for each , a function as follows:
Now, we are ready for the next condition:
if , then ,
if , then for .
Note that if were a complete Boolean algebra, we can take to be , the set of all functions from to .
Remarks. Let be the functional polyadic algebra for .
is a polyadic algebra. The proof of this is not difficult, but involved, and can be found in the reference below.
If is a singleton, then can be identified with the functional monadic algebra for , for is just , and is just .
If is , then can be identified with the Boolean algebra , for and is a singleton, and hence the set of functions from to is identified with .
- 1 P. Halmos, Algebraic Logic, Chelsea Publishing Co. New York (1962).
|Title||example of polyadic algebra|
|Date of creation||2013-03-22 17:53:20|
|Last modified on||2013-03-22 17:53:20|
|Last modified by||CWoo (3771)|
|Defines||functional polyadic algebra|