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

TL;DR

This paper explores the atomic structure of C-algebras, characterizing atoms, conditions for atomicity, and using if-then-else actions to classify elements, advancing understanding of their algebraic properties.

Contribution

It introduces atomicity concepts in C-algebras, characterizes atoms in transformation algebras, and links finite atomic C-algebras to finite adas.

Findings

Atoms characterized in C-algebras of transformations

Finite atomic C-algebras are exactly finite adas

Conditions for atomicity established

Abstract

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.