Cancellation Meadows: A Generic Basis Theorem and Some Applications

Abstract

Let ℚ₀ denote the rational numbers expanded to a ‘meadow’, that is, after taking its zero-totalized form (0⁻¹=0) as the preferred interpretation. In this paper, we consider ‘cancellation meadows’, i.e. meadows without proper zero divisors, such as ℚ₀ and prove a generic completeness result. We apply this result to cancellation meadows expanded with differentiation operators, the sign function, and with floor, ceiling and a signed variant of the square root, respectively. We give an equational axiomatization of these operators and thus obtain a finite basis for various expanded cancellation meadows.