TY - JOUR
AU - Bergstra, Jan Aldert
AU - Tucker, John V.
PY - 2022/03/28
Y2 - 2023/01/29
TI - Totalising Partial Algebras: Teams and Splinters
JF - Transmathematica
JA - Transmath
VL -
IS - 0
SE - Primary Article
DO - 10.36285/tm.57
UR - https://transmathematica.org/index.php/journal/article/view/57
SP -
AB - <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 < 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>
ER -