Arithmetical meadows

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.

💡 Research Summary

The paper revisits the algebraic structure known as a meadow—a commutative ring equipped with a total multiplicative inverse operation defined by assigning the value 0 to the inverse of 0. Traditional meadows (sometimes called “inversive meadows”) extend the usual field axioms by making division by zero a well‑defined operation, thereby avoiding partial functions. Building on this foundation, the authors adopt Peacock’s tradition of “arithmetical algebra” and propose several variants that deliberately omit an additive identity (0) and an additive inverse (‑). In other words, the addition operator remains, but the special element that normally serves as the neutral element for addition is removed, and consequently the notion of subtraction is absent.

Two main families of structures are introduced. The first, called an inversive arithmetical meadow, retains the total inverse operation of a standard meadow but works in a setting without a designated additive zero. The second, a divisive arithmetical meadow, replaces the inverse operation with a total division operator “÷”. In both cases the rule “a ÷ 0 = 0” (or equivalently “0⁻¹ = 0”) guarantees that division is a total function. The multiplicative identity 1 is retained, and the usual ring axioms for addition (commutativity, associativity) and multiplication (commutativity, associativity, distributivity) are kept, except for any axioms that explicitly involve the additive zero or additive inverses.

The authors present a complete equational axiomatization for each class. The axiom sets consist of three blocks:

1. Ring‑core axioms – commutative, associative, distributive laws, and the existence of a multiplicative unit 1.
2. Total inverse/division axioms – equations that capture the behaviour of the inverse or division on non‑zero elements while forcing the result to be 0 when the divisor is 0. A typical formulation uses a conditional multiplication operator:
   a·a⁻¹ = 1·(a ≠ 0) + 0·(a = 0).
   In the divisive setting the same effect is expressed by a ÷ b = a·b⁻¹.
3. Addition‑only axioms – associativity and commutativity of addition are retained, but any identity law involving 0 (e.g., a + 0 = a) is omitted, reflecting the intentional absence of an additive neutral element.

The paper proves that these axiom systems are sound (all derivable equations hold in the intended models) and complete (every true equation in the models can be derived from the axioms). Completeness is established by constructing initial algebras for each theory and by demonstrating that standard number systems—natural numbers, integers, and real numbers—augmented with the rule “0⁻¹ = 0” satisfy the axioms. The authors also discuss how the conditional operator can be eliminated in favour of ordinary equational reasoning by introducing auxiliary symbols that encode the “is‑zero” test.

Concrete instances are provided to illustrate the theories. For example, the set of natural numbers ℕ equipped with the usual operations and the definition 0⁻¹ = 0 forms an inversive arithmetical meadow. Similarly, the real numbers ℝ with the same rule become a divisive arithmetical meadow; here division by zero yields 0, which preserves most analytic properties while simplifying error handling. A more algebraic example is the polynomial ring ℤ