Evidently, the notions of "being dimensionful" and "being dimensional homogeneous" are topics of consideration related to (certain) physical quantities, their values, and their value ranges.
I'm interested in capturing these notions in a mathematically unambiguous sense; and therefore, case by case, I'm considering a suitable non-empty set, say $\mathcal V$, with the following algebraic properties (expressed in terms of the product $(\mathbb R * \mathcal V)$, and the comparison relation $ (=) $ defined between any two of those products):
- that each element is "dimensionful", i.e. "not just some (real) number":
$$ \mathcal V \cap \mathbb R = \emptyset, \tag{1}, $$
- that all elements are jointly "dimensional homogeneous" in the sense of being "commensurate (with respect to real numbers)" to each other; separately, but consistently, for a null-element $\mathbf z$ (if there is one in set $\mathcal V$):
$$ (\exists \, \mathbf z \in \mathcal V : 1 * \mathbf z = 0 * \mathbf z) \implies (\forall \, \mathbf a \in \mathcal V : 1 * \mathbf z = 0 * \mathbf a) \tag{2}, $$
and separately, but consistently, for all other elements of $\mathcal V$ (if there are any):
$$ \forall \, \mathbf a \in \mathcal V : (1 * \mathbf a \ne 0 * \mathbf a) \implies (\forall \, \mathbf b \in \mathcal V : \exists \, s_{ba} \in \mathbb R \, | \, 1 * \mathbf b = s_{ba} * \mathbf a) \tag{3},$$
and finally that
- the elements are each unique (with respect to the above "commensurability", or in other words: "by value"):
$$\forall \, \mathbf a, \mathbf b \in \mathcal V : (1 * \mathbf a = 1 * \mathbf b) \implies (\mathbf a \equiv \mathbf b) \tag{4}.$$
Note that one-dimensional vector spaces conform to properties $(1 - 4)$ above, but they do have additional properties; e.g. they each must include a null-element, the so-called zero vector; and they must include inverse ("negative") elements with respect to vector addition.
My question:
Does the system of algebraic properties $(1 - 4)$ as presented above, applicable to the elements of one suitable given set $\mathcal V$, already have a specific name ?
(Obviously, it could not be called "the algebraic properites of a one-dimensional vector space", since those are "stronger", involving the additional properties noted above.)
Have these exact properties $(1 - 4)$ been explictly considered elsewhere ?