Fields and Meadows

This is work together with John Tucker (Swansea) on the application of EAS (elementary algebraic specifications) to the theory of fields. We have designed an equational specification ENA (elementary number algebra) which specifies a super class of the class of zero-totalized fields. Models of ENA are called meadows. An equivalent subset of 10 equations has been termed Md in a joint paper with Yoram Hirschfeld (Tel Aviv) and John Tucker simply called Meadows. Every field can be expanded to a meadow but not conversely. Dutch presentation contains a slideshow in Dutch with an easygoing introduction to the subject. An invited morning lecture has been presented at BMC 2007 in Swansea. The results of this activity that have been obtained until now are these:

• An EAS specification: ENA+L has been given for the zero totalised rational numbers. Moreover Ground completeness of ENA for the (first order) theory of zero-totalised fields. These results have been published in a J. ACM paper of April 2007 by JAB and JVT, carrying the title: `The rational numbers as an abstract data type'.
• JAB, YH and JVT have extended the ground completeness result to a completeness results that holds for all equations in Meadows. In that paper we also characterize meadows as expansions of commutative von Neumann regular rings following a suggestion made by Robin Chapman (Exeter) during BMC2007.
• An EAS specification for the zero totalised complex rationals (Festschrift 'Joe Goguen' by JAB and JVT)
• An EAS for the zero totalised field of rational functions (invited lecture, CIE 2006 by JAB)
• A characterization of finite minimal meadows has been obtained (unpublished).
• A significant collection of open research problems has been accumulated.
• Three different methods have been identified for division safe calculation in zero totalised fields.

• Inge Bethke and Piet Rodenburg (UvA) are performing calculations concerning the structure of finite meadows.
• JAB and JVT investigate an adaptation of zero totalized fields to include the additional and more complicated feature of error totalized fields in an attempt to formalize phenomena related to the FIT-C, a calculator that has been designed by Harald and William Thimbleby (Swansea).
• JAB works with Sanne Nolst Trenite and Mark van der Zwaag (both UvA) on a formalization of of financial bugets which combines module algebra with the equational logic of zero totalized fields.