On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness

Common meadows are arithmetic structures with inverse or division, made total on $0$ by a flag $\bot $ for ease of calculation. We examine some axiomatizations of common meadows to clarify their relationship with commutative rings and serve different theoretical agendas. A common meadow fracterm calculus is a special form of the equational axiomatization of common meadows, originally based on the use of division on the rational numbers. We study axioms that allow the basic process of simplifying complex expressions involving division. A useful axiomatic extension of the common meadow fracterm calculus imposes the requirement that the characteristic of common meadows be zero (using a simple infinite scheme of closed equations). It is known that these axioms are complete for the full equational theory of common cancellation meadows of characteristic $0$. Here, we show that these axioms do not prove all conditional equations which hold in all common cancellation meadows of characteristic $0$.