Journal article 3 views
Fracterm Calculus for Signed Common Meadows
John Tucker 
TRANSMATHEMATICA, Volume: 2024
Swansea University Author:
John Tucker
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DOI (Published version): https://doi.org/10.36285/tm.97
Abstract
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 lim...
| Published in: | TRANSMATHEMATICA |
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| Published: |
2024
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa66862 |
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| Abstract: |
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 ofcomplex numbers with a weak sign function. We show that all common meadows may be enlarged to a weakly signed common meadow. A special case are 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. |
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| Keywords: |
quational calculus, fracterm calculus, ordered fields, common meadow, sign function, equational specification, initial algebra semantics |
| College: |
Faculty of Science and Engineering |