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In order to study the axiomatization of the if-then-else construct overpossibly non-halting programs and tests, the notion of $C$-sets was introducedin the literature by considering the tests from an abstract $C$-algebra. Thispaper extends the notion of $C$-sets to $C$-monoids which include thecomposition of programs as well as composition of programs with tests. For theclass of $C$-monoids where the $C$-algebras are adas a canonical representationin terms of functional $C$-monoids is obtained.

Axiomatization of if-then-else over monoids of possibly non-halting programs and tests