# abstract algebra for component wise operations on “vectors” or what it might be called

I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations: - multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied by w=(w1, w2, w3,...) gives v.w = (v1 w1, v2 w2, v3 w3,...) - no scalar multiplication of vectors allowed - all mathematical operators suc as log, sin, cos, sign, etc are applied componentwise - the vectors contain only integers In general every operation seems to work here component wise. Indeed I do not even like to call them even a vector. Formally, I need to fix what type of ring/module/... and properties define exactly this algebra and what terminology I can use. It is crazy that I have solved the problem but can not expressit in a formal language, because of lacking as from natural sciences the mathematical formalism from abstract algebra. Is there anybody who can help me please. Many thanks -Sam

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Since all of the operations are one component at a time, the resulting equations just involve separate copies of the real number line (or whatever ring your scalars live in), so it is just usual algebra of real numbers (or whatever ring your scalars live in). – Ben McKay Mar 18 '13 at 15:29
An extremely similar question was asked here before: mathoverflow.net/questions/9166. My preferred to think of $n$-vectors in this context as real-valued functions on a discrete space with $n$ points. – Igor Khavkine Mar 18 '13 at 15:44
Perhaps looking up "Cartsian product" or "direct product" of algebras will help. My guess is that the algebra you want is sometimes called $R \times R$ . Gerhard "Ask Me About General Algebra" Paseman, 2013.03.18 – Gerhard Paseman Mar 18 '13 at 15:45
Sorry: Cartesian. Gerhard "Smartphone Keyboards Are Too Small" Paseman, 2013.03.18 – Gerhard Paseman Mar 18 '13 at 15:47
How can the vectors only contain integers, yet you want to take $\sin$? – Allen Knutson Mar 18 '13 at 15:53