Arithmetical meadows

TL;DR

This paper introduces variants of inversive and divisive meadows, algebraic structures that extend rings with totalized inverse or division operations, and provides their equational axiomatizations.

Contribution

It defines and axiomatizes new variants of inversive and divisive meadows without additive identities or inverses, expanding the algebraic framework.

Findings

Axiomatizations of several classes of these variants

Identification of instances of these algebraic structures

Extension of arithmetical algebra concepts

Abstract

An inversive meadow is a commutative ring with identity equipped with a multiplicative inverse operation made total by choosing 0 as its value at 0. Previously, inversive meadows were shortly called meadows. A divisive meadow is an inversive meadows with the multiplicative inverse operation replaced by a division operation. In the spirit of Peacock's arithmetical algebra, we introduce variants of inversive and divisive meadows without an additive identity element and an additive inverse operation. We give equational axiomatizations of several classes of such variants of inversive and divisive meadows as well as of several instances of them.