4 ms·

I'd like to seen an extension of this table with the negation of these axioms

by 7373737373 7y ago

hope-striker 7y ago
What would you do with that? For example, I can see the use of commutativity (ab = ba) and anticommutativity (ab = -ba), but I'm not sure what I'd do with the negation of commutativity (ab ≠ ba).
- pfortuny 7y ago
  Nope: the negation is "there is a couple a,b such that ab!=ba", which means just "strictly not commutative group": I do not think there is a relevant theory to be done about them (otherwise, I guess it would have been done).
  - hope-striker 7y ago
    Ah, whoops. That seems even more useless, though.
- mikhailfranco 7y ago
  Non-commutative also means that if there are units, there may be different left and right units: 1L * A = A = A * 1R
QuesnayJr 7y ago
I think negations usually don't prove interesting, and mathematicians essentially use "non-" to mean "not necessarily". So the theory of "noncommutative rings" includes the theory of "commutative rings" as an easy special case.
deleted 7y ago
[deleted]
mikhailfranco 7y ago
Here is an beautiful old paper about data structures built using a binary join operator, exploring the 16 possible outcomes for properties: unit, idempotent, associative, commutative. The resulting grid can be factored around set, bag, list and binary tree, with empty/non-empty variants. Then there is interaction of the structures with binary operators on the data elements themselves, giving a nice analysis of map, filter, fold (reduce) in functional programming. A.Bunkenburg, The Boom Hierarchy http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.49.3252&rep=rep1&type=pdf http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.49.... There is a rich literature if you chase the references back and forth, starting from Bird-Meertens Formalism (Squiggol), Hoogendijk, through Backhouse and Malcolm, to Meijer and McBride.

4 ms·

I'd like to seen an extension of this table with the negation of these axioms

by 7373737373 7y ago

I'd like to seen an extension of this table with the negation of these axioms

hope-striker 7y ago
What would you do with that? For example, I can see the use of commutativity (ab = ba) and anticommutativity (ab = -ba), but I'm not sure what I'd do with the negation of commutativity (ab ≠ ba).
- pfortuny 7y ago
  Nope: the negation is "there is a couple a,b such that ab!=ba", which means just "strictly not commutative group": I do not think there is a relevant theory to be done about them (otherwise, I guess it would have been done).
  - hope-striker 7y ago
    Ah, whoops. That seems even more useless, though.
- mikhailfranco 7y ago
  Non-commutative also means that if there are units, there may be different left and right units: 1L * A = A = A * 1R
QuesnayJr 7y ago
I think negations usually don't prove interesting, and mathematicians essentially use "non-" to mean "not necessarily". So the theory of "noncommutative rings" includes the theory of "commutative rings" as an easy special case.
deleted 7y ago
[deleted]
mikhailfranco 7y ago
Here is an beautiful old paper about data structures built using a binary join operator, exploring the 16 possible outcomes for properties: unit, idempotent, associative, commutative. The resulting grid can be factored around set, bag, list and binary tree, with empty/non-empty variants. Then there is interaction of the structures with binary operators on the data elements themselves, giving a nice analysis of map, filter, fold (reduce) in functional programming. A.Bunkenburg, The Boom Hierarchy http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.49.3252&rep=rep1&type=pdf http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.49.... There is a rich literature if you chase the references back and forth, starting from Bird-Meertens Formalism (Squiggol), Hoogendijk, through Backhouse and Malcolm, to Meijer and McBride.