Abstract
We consider several novel congruences on the signature of meadows with the aim to survey different notions of fractions. In particular we suggest a notion of “true fraction”.
Highlights
This paper is written in honour of Rob van Glabbeek on the occasion of his 60th birthday.We congratulate Rob with this milestone and with his outstanding performance in the area of process theory.The idea of meadows is to introduce, on top of the signature of rings and fields, a function symbol for inverse, obtaining inversive notation x−1, or for division, thereby obtaining divisive notation (x/y, usually with the understanding that x−1 = 1/x), and to insist, or at least to prefer, that operations are total.An arithmetical datatype, admittedly an informal notion, is an abstract datatype which provides sorts and functions closely related to arithmetic.1Assuming that operations are total and that division is a named operation, the equational logic of known and sometimes novel structures is investigated
In [2] it is shown that fractions involving a single variable can be brought in so-called mixed fraction format
Some terminology which is conventional in the area of fractions has not been introduced in Sect
Summary
This paper is written in honour of Rob van Glabbeek on the occasion of his 60th birthday. The idea of meadows is to introduce, on top of the signature of rings and fields, a function symbol for inverse, obtaining inversive notation x−1, or for division, thereby obtaining divisive notation (x/y, usually with the understanding that x−1 = 1/x), and to insist, or at least to prefer, that operations are total. For equational axioms for meadows see [5,11,21,23] and for further theoretical information we refer to [3,4].2
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