If-then-else over the algebra of conditional logic

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This thesis aims at giving an axiomatization for the operation of if-then-else over algebras of non-halting programs and non-halting tests, and further, makes use of this axiomatization to study structural properties of the algebra of conditional logic. To this aim the thesis introduces the notion of C-sets by considering the tests from a C-algebra. When the Calgebra is an ada, the axiomatization is shown to be complete through a subdirect representation. Further, this thesis gives an axiomatization for the equality test along with if-then-else through the notion of agreeable C-sets, which is complete for the class of agreeable C-sets where the C-algebra is an ada. The thesis also introduces the notion of C-monoids which consider the composition of programs as well as composition of programs with tests along with if-then-else. A Cayley-type theorem is obtained in that every C-monoid where the C-algebra is an ada is embeddable in a functional C-monoid.The thesis also uses the if-then-else action to study the structure of C-algebras through the notions of annihilators and idempotence, through which a classification of elements of the C-algebra of transformations 3X is achieved. The thesis also proposes the notions of atoms and atomicity in C-algebras and obtains a characterisation of atoms in 3X. Further, the thesis presents necessary or sufficient conditions for the atomicity of C-algebras and shows that the class of finite atomic C-algebras is precisely that of finite adas.
Supervisors: K. V. Krishna & Purandar Bhaduri