One-dimensional vector spaces without a norm or any "further structure" are of course sets (in the following e.g. set $\mathcal V$) with some rather basic suitable properties; surely involving suitable "algebraic structure" consisting of multiplication with real numbers and comparison (equality, or inequality) of resulting products. Is the following definition of a (any "real") one-dimensional vector space correct and complete ? :
- Its non-null elements are all "commensurate (with respect to real numbers)" to each other:
$$\forall \, \mathbf a \in \mathcal V : (1 * \mathbf a \ne 0 * \mathbf a) \implies (\forall \, \mathbf b \in \mathcal V : \exists \, s_{ba} \in \mathbb R \, | \, 1 * \mathbf b = s_{ba} * \mathbf a),$$
- and its elements are each unique (with respect to the above "commensurability"):
$$\forall \, \mathbf a, \mathbf b \in \mathcal V : (1 * \mathbf a = 1 * \mathbf b) \implies (\mathbf a \equiv \mathbf b).$$
My question:
Are such basic properties and algebraic structure described and discussed more abstractly in the mathematics literature, perhaps using different terminology ?
Also, might those basic one-dimensional vector spaces without "further structure" perhaps be commonly known by some simpler, shorther name -- such as "commensurate spaces" ?
Note: I am interested in the described properties of one-dimensional vector spaces without a norm or any "further structure" as mathematical representation of the so-called "commensurability of values of the same physical quantity" (and "incommensurability of values of distinct physical quantities"); i.e. not necessarily in so-called "mathematical commensurability".