https://transmathematica.org/index.php/journal/issue/feed 2024-06-14T07:54:02-07:00 Dr James A. D. W. Anderson james.a.d.w.anderson@btinternet.com Open Journal Systems <p>Publishes articles and digital works in the sciences, humanities and arts of total (exception free) systems and new or adventurous mathematics.</p> https://transmathematica.org/index.php/journal/article/view/97 2024-04-08T02:07:08-07:00 Jan Aldert Bergstra j.a.bergstra@uva.nl John V Tucker j.v.tucker@swansea.ac.u

<p>A common meadow is an enrichment of a field with a division operator and an error value to make division total. A signed common meadow enriches a common meadow with a sign function that can be equationally axiomatised; the sign function can simulate an ordering on the underlying field but is not limited to orderings. In particular, of mathematical interest are the weakly signed common meadows. The prime example of a weakly signed common meadow is an expansion of a common meadow of complex numbers with a weak sign function. We show that all common meadows may be enlarged to a weakly signed common meadow. A special case is the 4-signed common meadows, which are precisely the enlargements of ordered fields. To illustrate the equational calculus for signed common meadows, we use it as a foundation for building a probability calculus and derive some classical formulae.</p>

2024-06-14T00:00:00-07:00 Copyright (c) 2024 Jan Aldert Bergstra, John V Tucker