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

This paper investigates the notions of atoms and atomicity in C-algebras and obtains a characterisation of atoms in the C-algebra of transformations. In this connection, various characterisations for the existence of suprema of subsets of C-algebras are obtained. Further, this work presents some necessary conditions and some sufficient conditions for the atomicity of C-algebras and shows that the class of finite atomic C-algebras is precisely the class of finite adas. This paper also uses the intrinsic if-then-else action to study the structure of C-algebras and classify the elements of the C-algebra of transformations.