Vector spaces without fields

In mathematics, the conventional definition of the concept of vector space relies upon the algebraic concept of a field. This article treats an algebraic definition that does not require that concept. If vector spaces are redefined as (universal) algebras as below, no preliminary introduction of fields is necessary. On the contrary, fields will come from such vector space algebras.

One of the ways to do this is to keep the first four abelian group axioms on addition in the standard formal definition and to formalize its geometric idea of scaling only by notions that concern every universal algebra.^[1] Vector space algebras consist of one binary operation "+" and of unary operations δ, which form a nonempty set Δ, that satisfy the following conditions, which do not involve fields.

1. (Total homogeneous algebra) There is a single set V such that every operation takes its two arguments or its argument from the whole V and gives its value in it.
2. (Abelian group) + satisfies the above mentioned axioms.
3. (Basis dilation) There is a basis set $B\subseteq V$ such that, for every such δ that is not constant, all the values $\delta(b)$, where b ranges over B, again form a basis set.
4. (Dilatations) Δ is the set of functions δ that satisfy the previous conditions and preserve all operations, namely $\delta(v+w)=\delta(v)+\delta(w)$ and $\delta(\gamma(v))=\gamma(\delta(v))$, for all $\gamma\in\Delta$ and all $v,w\in V$.

Ricci (2008) proves that these vector space algebras are the very universal algebras that any standard vector space defines by its addition and the scalar multiplications by any given scalar (namely each $a\in F$ gets a $\delta\in\Delta$ such that $\delta(v)=av$). Ricci (2007) proves that every such a universal algebra defines a suitable field. (Hence, it proves that these conditions imply all the axioms of the standard formal definition, as well as all the defining properties in definition 3 of a field.)^[2]

Since the field is defined from such vector space algebra, this is an algebraic construction of fields, which is an instance of a more general algebraic construction: the "endowed dilatation monoid" (Ricci 2007). However, as far as fields are concerned, there also is a geometric construction. Chapter II in (Artin 1957) shows how to get them starting from geometric axioms concerning points and lines.

Notes

1. The “generalized conception of space” in the preface of Whitehead (1898) is the first published claim that geometric ideas concern Universal Algebra too. It expands a similar claim in the preface of Fibonacci (1202).
2. This construction also concern the space with only one (null) vector, where the defined field only has the zero as element. This slightly extends definition 3 of a field, which only considers fields with at least two elements.

References

1. Artin, E. (1957). Geometric Algebra. New York: Interscience Publishers.
2. Fibonacci, L. (1857) [1202]. Liber Abbaci. Rome: reprinted by Tipografia delle Scienze Matematiche e Fisiche, B. Boncompagni ed.
3. Ricci, G. (2007). "Dilatations kill fields". International Journal of Mathematics, Game Theory and Algebra 16 (5/6): 13–34.
4. Ricci, G. (April 2008). "Another characterization of vector spaces without fields". In G. Dorfer, G. Eigenthaler, H. Kautschitsch, W. More, W.B. Müller (Hrsg.). Contributions to General Algebra 18. Klagenfurt: Verlag Johannes Heyn GmbH & Co. pp. 139–150. ISBN 978-3-7084-0303-8.
5. Whitehead, A.N. (1898). A treatise on Universal Algebra with applications 1. Cambridge: Cambridge University Press.