Maths/CogSci/MathPsych lazyweb: Are there any algebras in which you have subtraction but don't have negative values? Pointers appreciated. I am hoping that the abstract maths might shed some light on a problem in cognitive modelling.
The context is that I am interested in formal models of cognitive representations and I want to represent things (e.g. cats), don't believe that we should be able to represent negated things (i.e. I don't think it should be able to represent anti-cats), but it makes sense to subtract representations (e.g. remove the representation of a cat from the representation of a cat and a dog, leaving only the representation of the dog).
> do you mean something like an "'additive inverse'
> in an abstract algebra 'ring'"?
Quite possibly. My maths skills are vanishingly small. I'm interested in a problem domain that sounds like it might be modelled by a mathematician as some type of algebra.
What I'm hoping is that somebody says "That's a blah algebra". Mathematicians really care about internal consistency and the ultimate implications of things, so I would expect they would have thoroughly kicked the tyres of blah algebras and peered into all the nooks and crannies. Then, to the extent that I can wrap my head around the maths literature on blah algebra, I would hope to learn some things that might apply back in my original problem domain.
@RossGayler@cogsci a ring has some of the properties you are looking for, and it's a very general structure that has been well beaten on by mathematicians, but I simply dont know from the description here if its what you are looking for. Modulo arithmetic is an example of a ring, but a ring can be defined over many kinds of elements (such as matrices).
Suppose we have a cat as an element of a ring. Suppose we have a dog as an element of a ring. Adding those together gives you a cat and a dog which will also be an element of the ring. We won't have an "anti-dog" as something that makes sense to think about independently as just the negation of dog, but there will be an element of the ring that, if you add it to dog, will bring you back to nothing.
@RossGayler@Heterokromia@cogsci to me it seems you need to be more clear on your requirements. Are your non-negative and multidimensional requirements independent, as far as you can tell?
If so, a multidimensional (do you know how many dimensions/animals you have?) modulo space sounds a viable solution. That'd be something denoted as https://www.HostMath.com/Show.aspx?Code=%5Cmathbb%7BZ%7D_k%5En , with k being the cardinality of one dimension (would they need to have different cardinalities?), and n being the number of dimensions.