On the structure of C-algebras through atomicity and if-then-else

This paper introduces the notions of atoms and atomicity in $C$-algebras and obtains a characterisation of atoms in the $C$-algebra of transformations. Further, this work presents some necessary conditions and sufficient conditions for the atomicity of $C$-algebras and shows that the class of finite atomic $C$-algebras is precisely that of finite adas. This paper also uses the if-then-else action to study the structure of $C$-algebras and classify the elements of the $C$-algebra of transformations.