@article{Bergstra_Tucker_2022, title={Totalising Partial Algebras: Teams and Splinters}, url={https://transmathematica.org/index.php/journal/article/view/57}, DOI={10.36285/tm.57}, abstractNote={<p>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 &lt; m.<span class="Apple-converted-space"> </span>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.</p>}, journal={Transmathematica}, author={Bergstra, Jan Aldert and Tucker, John V.}, year={2022}, month={Mar.} }