Boolean Algebras as Clones

Abstract: Clones are many-sorted algebraic structures abstracting the composition of finitary operations, and play a central role in universal algebra and theoretical computer science. In this paper, we investigate the variety of clones generated by the clone whose only elements are the projections. Inspired by the algebraic studies of conditional statements in programming, we provide an equational axiomatisation of this class of clones, which we call partition clones. We prove that every partition clone is a subdirect product of the clone of projections. Partition clones arise from Boolean algebras via a natural construction, and we show that every partition clone can be actually constructed in this way. Finally, we investigate the algebras associated with partition clones; we prove that they correspond precisely to sets equipped with an action of a Boolean algebra, aligning with a well-known definition due to Bergman.