Cancellation Meadows: a Generic Basis Theorem and Some Applications

TL;DR

This paper establishes a generic completeness theorem for cancellation meadows, a class of algebraic structures, and applies it to various expanded meadows with additional operators, providing finite axiomatizations.

Contribution

It introduces a generic basis theorem for cancellation meadows and derives finite axiomatizations for meadows expanded with differentiation, sign, floor, ceiling, and square root functions.

Findings

Proves a generic completeness result for cancellation meadows.

Provides finite axiomatizations for meadows with additional operators.

Applies the theorem to various operator-extended meadows.

Abstract

Let Q_0 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 $Q_0$ 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.