Quantifiers for monoids

["Uday S. Reddy" <u-reddy@cs.uiuc.edu>]

In studying Algol-like languages, I repeatedly run into operators that
have an interesting structure.  I am wondering if such operators are
studied somewhere.

Consider a monoid <M,*,1> in a CCC.  The operations of interest are
natural transformations E_A : [A => M] -> M that satisfy the following
equations (in the internal language of the CCC):

    E_A(\lambda x. 1)  =  1
    E_A(\lambda x. a * g`x) = a * E_A(g)
    E_A(\lambda x. g`x * a) = E_A(g) * a
    E_A(\lambda x. E_B(\lambda y. h`x`y)) = 
			E_B(\lambda y. E_A(\lambda x. h`x`y))

These operators "feel" like existential quantifiers.  In fact, if M is
a subobject classifier with the monoid structure ofh conjunction, then
the existential quantifier E satisfies all of these equations (though
it is not a natural transformation).

In the applications I am interested in, M is a type of commands, with
* as sequential composition and 1 as the null action.  An example of
E is a local variable declaration.  

Is there some algebra or theory related to these kinds of operators?

Cheers,
Uday Reddy