example of monadic algebra
defined above is a Boolean subalgebra of .
We need to show that, (1): , (2): for any , , and (3): for any , .
Suppose . Then . By de Morgan’s law on infinite joins, the last expression is , which exists. Dually, exists by de Morgan’s law on infinite meets. Therefore, .
The three conditions are verified and the proof is complete. ∎
Remark. Every constant function belongs to .
For each , write and . Define two functions by
Since these are constant functions, they belong to .
Now, we define operators on by setting
By the remark above, and are well-defined functions on ().
is an existential quantifier operator on and is its dual.
The following three conditions need to be verified:
Finally, to see that is the dual of , we do the following computations:
completing the proof. ∎
Based on Propositions 1 and 2, is a monadic algebra, and is called the functional monadic algebra for the pair .
|Title||example of monadic algebra|
|Date of creation||2013-03-22 17:51:55|
|Last modified on||2013-03-22 17:51:55|
|Last modified by||CWoo (3771)|
|Defines||functional monadic algebra|