Can multiplication and division be treated as logical operations? A few of my friends and I were playing around with math (more specifically, why (-1)(-1)=1) and we figured out that multiplication (with regards to signs) was an "nxor" operation (I.E. If we treat "1" as "true" and "-1" as "false," than the values of multiplication, again with regards to signs, are the same as the "nxor" operation.) Now, we've begun thinking about redefining multiplication as other logical operations (For example: under "and" (-1)(-1)=-1) My questions are these: is this line of thought similar to any current area of mathematical research? If so, where can I go to find more information on it. I am especially interested in any proven theorems or open conjectures on this topic.

" My questions are these: is this line of thought similar to any current area of mathematical research?"
Yes, absolutely. The area you rediscovered is called algebraic logic. I think it is not a very active area of research any more, but in 50's and 60's it was rather active. Especially Tarski school of logic did many things in this area. You may want to check this wkikipedia page https://en.wikipedia.org/wiki/Algebraic_logic
