Monoids of non-halting programs with tests

In order to study the axiomatization of the if-then-else construct over possibly non-halting programs and tests, the notion of C-sets was introduced in the literature by considering the tests from an abstract C-algebra. This paper extends the notion of C-sets to C-monoids which include the composition of programs as well as composition of programs with tests. For the class of C-monoids where the C-algebras are adas a canonical representation in terms of functional C-monoids is obtained.