Inversive Meadows and Divisive Meadows

TL;DR

This paper introduces finite equational specifications for inversive and divisive meadows, algebraic structures with totalized inverse and division operations, and explores their variants and foundational aspects in mathematics and computer science.

Contribution

It provides the first finite equational specifications for all inversive and divisive meadows, including variants without additive identity or inverse, and analyzes their foundational implications.

Findings

Initial algebras of rational numbers can be obtained from these specifications.

Variants without additive identity or inverse are constructed and analyzed.

Divisive meadows are shown to be more fundamental than inversive meadows.

Abstract

Inversive meadows are commutative rings with a multiplicative identity element and a total multiplicative inverse operation whose value at 0 is 0. Divisive meadows are inversive meadows with the multiplicative inverse operation replaced by a division operation. We give finite equational specifications of the class of all inversive meadows and the class of all divisive meadows. It depends on the angle from which they are viewed whether inversive meadows or divisive meadows must be considered more basic. We show that inversive and divisive meadows of rational numbers can be obtained as initial algebras of finite equational specifications. In the spirit of Peacock's arithmetical algebra, we study variants of inversive and divisive meadows without an additive identity element and/or an additive inverse operation. We propose simple constructions of variants of inversive and divisive meadows with a partial multiplicative inverse or division operation from inversive and divisive meadows. Divisive meadows are more basic if these variants are considered as well. We give a simple account of how mathematicians deal with 1 / 0, in which meadows and a customary convention among mathematicians play prominent parts, and we make plausible that a convincing account, starting from the popular computer science viewpoint that 1 / 0 is undefined, by means of some logic of partial functions is not attainable.