[FOM] On 0^{-1}=0

[John Tucker]

I’d like to follow up various posts on 0^{-1}=0.

If you want to avoid the partial nature of division for some reason then there are many options. In this post I’ll speak of one that has been mentioned already: meadows.

1. In computer science the algebraic theory of data types provides a beautiful general theory of data based on many-sorted algebras and axioms that are, or are close to, equations.  The theory started as formal theory of specification. Here is one basic problem:  suppose data and its operations and tests are modelled as a many sorted algebra A with signature S, then find equations E, possibly expanding S using extra operators S’, such that a S-reduct of the initial algebra I(S’, E) is isomorphic to A.

2. The theory’s beauty is diminished somewhat if operations are partial, as was observed in the 1970s when attempts to axiomatise the stack met the operation of popping the empty stack. If only the pop operation could be total. See for example:

Bergstra and Tucker, The data type variety of stack algebras 
Annals of Pure and Applied Logic 73 (1995) 11-36
http://www.sciencedirect.com/science/article/pii/0168007294000385 <http://www.sciencedirect.com/science/article/pii/0168007294000385>


3. Given the theoretical importance of the rational numbers in computation — being the data type of finite representations of reals and physical measurements relative to a system of units — the problem of finding an equational specification of the rational numbers is interesting and challenging.  A successful attack on the problem was made in:

Bergstra and Tucker, The rational numbers as an abstract data type
Journal ACM 54 (19) Article 7 25 pages
DOI = 10.1145/1219092.1219095 

In order to make an equational specification of the rationals,  the question of totalising division was addressed and idea of a ‘generalisation’ of the concept of field to a meadow arose.


4. Meadows are a class of structures, with total operations of +, x , - and ^{-1}, that is equationally definable. The equations imply that 0^{-1}=0. Later, in the study

Bergstra, Hirshfeld and Tucker, Meadows and the equational specification of division
Theoretical Computer Science 410 (2009) 1261-1271
http://www.sciencedirect.com/science/article/pii/S0304397508008967 <http://www.sciencedirect.com/science/article/pii/S0304397508008967>

There were a number of theorems that demonstrated the concept had an interesting theory and quite some potential for further developments. For example, 

Theorem: Up to isomorphism, non-trivial meadows are precisely the subalgefbras of products of zero-totalised fields.

Theorem: The equational theory of meadows and the equational theory of fields with zero-totalised division are identical.


5. Subsequently, Jan Bergstra and his colleagues at Amsterdam have developed a good deal of the algebraic theory of meadows and some applications. A convenient source of references is his page on the CS Bibliography:

http://dblp.uni-trier.de/pers/hd/b/Bergstra:Jan_A=.html <http://dblp.uni-trier.de/pers/hd/b/Bergstra:Jan_A=.html>

A particularly interesting application is:

Meadow based fraction theory
http://arxiv.org/abs/1508.01334 <http://arxiv.org/abs/1508.01334>

There is much to say on the topic, most of which can be found in these papers. Total division makes sense algebraically and logically.

6. Given their algebraic theory is solid, are there precedents in the vast literature of abstract algebra? Meadows are close to the well-known von Neumann regular rings, expanded with a pseudo-inverse. However, the idea of 0^{-1}=0 is used in the idea of desirable pseudo fields: 

Komori, Free algebras over all fields and pseudofileds, 
Reports of the  Faculty of Science, Shizuoka University 10 (1975), 9-15
http://ir.lib.shizuoka.ac.jp/bitstream/10297/8858/1/10-0009.pdf

Ono, Equational theories and universal theories of fields
Journal of the Mathematical Society of Japan 356 (1983) 289-306
https://projecteuclid.org/download/pdf_1/euclid.jmsj/1230396550

7. As to other methods, there is the notion of wheel proposed by my colleague Anton Setzer in:

Setzer,Wheels, 1997. 
http://www.cs.swan.ac.uk/~csetzer/articles/wheel.pdf <http://www.cs.swan.ac.uk/~csetzer/articles/wheel.pdf>

The topic was developed in:

J Carlstrom, Wheels - on division by zero
Mathematical Structures in Computer Science 14 (2004) 143-184
http://journals.cambridge.org/action/displayFulltext?type=1&fid=198510&jid=MSC&volumeId=14&issueId=01&aid=198509

where wheels are equationally defined.


Formalisations like 0^{-1}=0 can reveal the power of taking syntax seriously. This is something that computer science has long understood and institutionalised in its culture, theory and practice, but mathematics remains reluctant to acknowledge outside logic.



John 








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