Totalising Partial Algebras

Teams and Splinters


  • Jan Aldert Bergstra Informatics Institute, University of Amsterdam
  • John V. Tucker Department of Computer Science, Swansea University



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.




How to Cite

Bergstra, J. A., & Tucker, J. V. (2022). Totalising Partial Algebras: Teams and Splinters. Transmathematica.



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