The primary mathematical structure for measurement and computation is unquestionably a field. Because fields do not have a purely equational axiomatization, the axioms of a field cannot be used in applications of the theory of abstract data types to number systems based on rational, real or complex numbers. This state of affairs brought Bergstra and Tucker in 2005 to introduce meadows as alternatives for fields with a purely equational axiomatization.
At the basis of meadows lies the decision to make the multiplicative inverse operation total by imposing that the multiplicative inverse of zero is zero. A meadow is a commutative ring with a multiplicative identity element and a total multiplicative inverse operation satisfying the two equations (x-1)-1 = x and x · (x · x-1) = x. It follows from the axioms of a meadow that the multiplicative inverse operation also satisfies the equation 0-1 = 0. All fields in which the multiplicative inverse of zero is zero, called zero-totalized fields, are meadows, but not conversely. Because of their purely equational axiomatization, all meadows are total algebras and the class of all meadows is a variety.
The work done on meadows so far provides among other things:
- equational specifications of the meadow of rational numbers under initial algebra semantics;
- complete equational axiomatizations of the equational theories of the meadow of real numbers and the meadow of complex numbers;
- typical properties of meadows, finite meadows, and initial meadows, including their relationship with fields;
- relationships between certain subvarieties of the variety of meadows;
- expansions of meadows with certain operations, such as the sign operation and the floor and ceiling operations;
- variants of meadows in which the multiplicative inverse operation is replaced by a division operation;
- variants of meadows in which multiplicative inverse of zero is different from zero (an additional value or not).
The applications of meadows done so far include applications in:
- tuplix calculus, a calculus that is concerned with transfers of quantities of something;
- an ACP-based process calculus that concerns processes in which quantities are involved;
- data linkage dynamics, a simple model of computation that pertains to the use of dynamic data structures and the calculation of quantities in programming;
- a rephrasing of Kolmogorov’s probability axioms for finitely additive probability spaces;
- probabilistic thread algebra, an algebraic theory of mathematical objects that represent the behaviours produced by probabilistic programs under execution;
- a precise account of fractions.
The general aim of this website is to bring meadows as alternatives for fields better into the picture. We try to achieve this by means of the following:
- an enumeration of the refereed papers and preprints in which the results of the work done on meadows so far is presented;
- an enumeration of the refereed papers and preprints in which the results of the work done on applications of meadows so far is presented;
- an enumeration of some open questions originating from the work done on meadows so far.
For refereed papers that are not freely available, a preprint or postprint is in most cases provided.
In closing, the following two remarks are in order. Firstly, meadows are precisely the commutative von Neumann regular rings with a multiplicative identity element expanded with an operation for taking weak inverses. Secondly, skew meadows, which differ from meadows only in that their multiplication operation is not required to be commutative, were already studied in 1975 by Komori, who named them desirable pseudo-fields.