  In order to study the axiomatization of the if-then-else construct over
possibly non-halting programs and tests, this paper introduces the notion of
$C$-sets by considering the tests from an abstract $C$-algebra. When the
$C$-algebra is an ada, the axiomatization is shown to be complete by obtaining
a subdirect representation of $C$-sets. Further, this paper considers the
equality test with the if-then-else construct and gives a complete
axiomatization through the notion of agreeable $C$-sets.
