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Totalising Partial Algebras

Jan Aldert Bergstra, John Tucker Orcid Logo

Transmathematica, Volume: 2022, Pages: 1 - 22

Swansea University Author: John Tucker Orcid Logo

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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...

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Published in: Transmathematica
ISSN: 2632-9212
Published: Reading Transmathematica 2022
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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.
Keywords: partiality, meadows, teams, splinters, absorptive elements
College: Faculty of Science and Engineering
Start Page: 1
End Page: 22