On Sat, Sep 12, 2015 at 5:12 AM, Jerzy Karczmarczuk
And whatever people say, e.g., Wren Romano:
There are far more objects which have addition/multiplication without subtraction than there are objects with addition/subtraction without multiplication.
... several simple-minded people (myself included) will find not so useful and/or clumsy the introduction of specific structures just to forbid the subtraction. Nobody will fight against these purists, it is simply a difficult issue.
I can assure you that there are numerous examples that even the most "simple-minded" programmers are already intimately familiar with; cf., http://winterkoninkje.dreamwidth.org/80410.html. People use tropical, arctic, and Viterbi semirings all over the place: e.g., almost[1] every minimization/maximization problem where there's some notion of gluing together subproblems. People use bounded distributed lattices all the time: e.g., most everything in naive set theory. People use regular languages/regexes all the time. All of these are semirings, and none of them are rings. Semirings are perfectly banal everyday things. The only difficulty is in realizing how ubiquitous they are. This has nothing to do with being a purist. I want semirings encoded as a type class because —as a workaday programmer— I find myself almost daily wanting to parameterize some algorithm to use a different semiring. All the structure of linear algebra is far more complicated than what's going on here. Most programmers ("simple-minded" or otherwise) do not actually work with linear algebra; but almost every programmer has to deal with sets, parsing/languages, minimizing/maximizing, not screwing up array indices by going negative, etc etc. [1] The salient min/max problems that don't fit that characterization are the ones where we have two notions of gluing together sub problems, in which case we generally want an ordered semiring— which is just a generalization. -- Live well, ~wren