## Commutative Unital Ring

Dawood, Hend, and Nefertiti Megahed. "

A Consistent and Categorical Axiomatization of Differentiation Arithmetic Applicable to First and Higher Order Derivatives."

*Punjab University Journal of Mathematics* 51, no. 11 (2019): 77-100.

AbstractDifferentiation arithmetic is a principal and accurate technique for the computational evaluation of derivatives of first and higher order. This article aims at recasting real differentiation arithmetic in a formalized theory of dyadic real differentiation numbers that provides a foundation for first and higher order automatic derivatives. After we set the stage by putting on a systematic basis certain fundamental notions of the algebra of differentiation numbers, we begin by setting up an axiomatic theory of real differentiation arithmetic, as a many-sorted extension of the theory of a continuously ordered field, and then establish the proofs for its consistency and categoricity. Next, we carefully construct the algebraic system of real differentiation arithmetic, deduce its fundamental properties, and prove that it constitutes a commutative unital ring. Furthermore, we describe briefly the extensionality of the system to an interval differentiation arithmetic and to an algebraically closed commutative ring of complex differentiation arithmetic. Finally, a word is said on machine realization of real differentiation arithmetic and its correctness, with an addendum on how to compute automatic derivatives of first and higher order.

Keywords: Automatic differentiation; Categorical differentiation arithmetic; Consistent differentiation arithmetic; Commutative unital ring; Interval differentiation arithmetic; Algebraically closed commutative rings.