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On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness
The Computer Journal, Volume: 66, Issue: 7
Swansea University Author:
John Tucker
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DOI (Published version): 10.1093/comjnl/bxac026
Abstract
Common meadows are arithmetic structures with inverse or division, made total on 0 by a flag⊥ for ease of calculation. We examine some axiomatizations of common meadows to clarify theirrelationship with commutative rings and serve different theoretical agendas. A common meadowfracterm calculus is a...
| Published in: | The Computer Journal |
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| ISSN: | 0010-4620 1460-2067 |
| Published: |
Oxford University Press (OUP)
2022
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| Online Access: |
Check full text
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa60587 |
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| Abstract: |
Common meadows are arithmetic structures with inverse or division, made total on 0 by a flag⊥ for ease of calculation. We examine some axiomatizations of common meadows to clarify theirrelationship with commutative rings and serve different theoretical agendas. A common meadowfracterm calculus is a special form of the equational axiomatization of common meadows, originallybased on the use of division on the rational numbers. We study axioms that allow the basic processof simplifying complex expressions involving division. A useful axiomatic extension of the commonmeadow fracterm calculus imposes the requirement that the characteristic of common meadows bezero (using a simple infinite scheme of closed equations). It is known that these axioms are completefor the full equational theory of common cancellation meadows of characteristic 0. Here, we showthat these axioms do not prove all conditional equations which hold in all common cancellationmeadows of characteristic 0. |
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| Keywords: |
Arithmetic structures; rational numbers; division by zero; meadows; common meadows; fracterm calculus; equational specification; initial algebra semantics |
| College: |
Faculty of Science and Engineering |
| Issue: |
7 |