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Totalising Partial Algebras
Transmathematica, Volume: 2022, Pages: 1 - 22
Swansea University Author:
John Tucker
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Copyright 2022 Jan Aldert Bergstra, John V. Tucker. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
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DOI (Published version): 10.36285/tm.57
Abstract
We will examine totalising a partial operation in a general algebra by using an absorbtive element, bottom, such as an error flag. We then focus on the simplest example of a partial operation, namely subtraction on the natural numbers: n - m is undefined whenever n < m. We examine the use of bott...
| Published in: | Transmathematica |
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| ISSN: | 2632-9212 |
| Published: |
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Transmathematica
2022
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa61536 |
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| Abstract: |
We will examine totalising a partial operation in a general algebra by using an absorbtive element, bottom, such as an error flag. We then focus on the simplest example of a partial operation, namely subtraction on the natural numbers: n - m is undefined whenever n < m. We examine the use of bottom in algebraic structures for the natural numbers, especially semigroups and semirings. We axiomatise this totalisation process and introduce the algebraic concept of a team, being an additive cancellative semigroup with totalised subtraction. Also, with the natural numbers in mind, we introduce the property of being generated by an iterative function, which we call a splinter. We prove a number of theorems about the algebraic specification of datatypes of natural numbers. |
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| Keywords: |
partiality, meadows, teams, splinters, absorptive elements |
| College: |
Faculty of Science and Engineering |
| Start Page: |
1 |
| End Page: |
22 |