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/sci/ - Science & Math

So I just started learning about Abstract Algebra and I'm curious to ask. Can you basically create your own algebraic structure such as a groups, rings, and fields? Like you make some arbitrary finite set and equip it with one or operators of any desired arity?
If i did this and if the structure becomes useful in some way could i write a paper about it?
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>>10718244
No. It doesn't have to be useful to write a paper about it. All you need is clear definitions. Never let your pure math be sullied by applicability to some problem domain.
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>>10718249
aaah I didnt think i would be advised by a pure math ubermensch such as yourself. I would do it for the sake of math itself. I was expecting engi bois to hurl feces at me.
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>nerd humour
*sigh*
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Good luck finding something that it both original and interesting.
It doesn't have to be applicable, but it does need to have something interesting about it.
Chances are you will be recreating some structure that somebody has already created and studied in the past.
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>>10718458
Well the axioms could match up to any sort of structure but the point is using different operations and a paticular set that might turn out to be something interesting like some operarion with Z/nZ for some n in N
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>>10718244
Yes, actually a decent amount of research is done in algebra (and logic) doing precisely this (although strictly speaking, this falls more so under "Universal Algebra" that "Abstract Algebra" just so you know). Often times we have an algebra (or logic) and we want to know what sorts of things it might be useful to describe. Similarly, we might have some sort of problem we need solved, and then we go on a search, so to speak and try to come up with some axioms and rules that would allow us to construct an algebra for analyzing the problem.

In fact their are dozens of different types of algebraic structure: groups, rings, fields, but also monoids, semigroups, modules, vector spaces, semirings/semimodules (not very common). Then we have various types of "algebra": froebenius, lie algebra, hopf algebra, etc. (these are mostly for physicists).

More along the lines of what you're thinking of are things like Heyting algebras. An interesting topic at the intersection of logic and algebra is finding, comparing, and contrasting model theoretic interpretations for certain types of logical system. For example Heyting Algebra provides a normal or typical algebraic semantics for modal logic. An alternative semantics can be provided by assigning various sorts of topological interpretation to the modal operators. Modal logic can then be seen a formal system or algebra for reasoning about topological properties.
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>>10719340
Thanks Anon for the clarification. I needed to hear this :)
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>>10718244
>Starting a sentence with an unnecessary "like".
Go like kill like yourself like, you like illiterate like basic like thot. Like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like like.

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