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Hey I have this question from Universal Algebra texts where you can see groups, rings, lattices and other structures as Universal Algebras, but I still don't have clear how vector spaces can be viewed in this way (taking into account that all the operations in an Universal Algebra are internal: i.e, from $A^n$ to $A$)



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2 Answers 2

Assuming you are looking at vector spaces over a fixed field $K$: you consider one binary operation $A\times A\to A$ for addition, one constant for the zero, one unary operation for additive inverses and one unary operation $m_\lambda:A\to A$ for multiplication by each scalar $\lambda\in K$. You end up with infinitely many operations if the field is infinite, of course.

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Mariano's answer is perfect, of course, but let me add a few more details and generalities.

You can view a ring with unit as a (universal) algebra $\langle R, +, \cdot, -, 0, 1\rangle$, and you can view a module over this ring as the algebra $\langle M, +, -, 0, \{f_r : r\in R\}\rangle$, where the reduct $\langle M, +, -, 0\rangle$ is an abelian group, each of the unary operations $f_r$ (scalar multiplication by $r$) is an endomorphism of this group, and the map $r \mapsto f_r$ is a ring homomorphism. If the ring $R$ happens to be a field, we call $M$ a vector space.

The best reference (imho) for this view of the world is "Algebras, Lattices, Varieties, Vol. 1" by McKenzie, McNulty, Taylor.

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