Let $S$ be a semifield, that is an algebraic structure satisfiying all field axioms except (perhaps) the existence of additive inverses. A semivector space is a semimodule (a module without additive inverses) over a semifield. Let $V$ be a semivector space over $S.$
I would like to understand what the definition of a basis of $V$ should be. I have found very little material about semivector spaces. All I have found is some papers by Indian author W. B. Vasantha Kandasamy. She defines a basis of $V$ by copying the definition of a basis of a vector space -- a tuple of vectors $v_1,...,v_n$ that generate the whole and are linearly independent, where "linearly independent" means
$\left(\forall a_1,...,a_n\in S\right) \;\;\left(\sum_{i=1}^{n}a_iv_i = 0 \Longrightarrow \left(\forall i\in\{1,...,n\}\right)\;\;a_i=0\right).$
I believe this is a very unnatural definition. I think that linear independence should be always understood as "freeness" of the spanned space. I know next to nothing about category theory but I understand a free space is one "with no non-trivial relations in it". The above condition doesn't assure that there are no non-trivial relations because we cannot use subtraction to move one side of a possible non-trivial relation to the other side.
I have two questions.
- Do you think I'm correct and Kandasamy's definition doesn't make much sense?
- If so, could you please help me build a correct definition of linear independence for semivector spaces? I think it's a very good exercise for me to improve my understanding of free objects. I've been trying to find a "relational" definition and then prove that it's equivalent to this defintion. But I'm not doing very well. Should I define "non-trivial relation" to be a formula of the form
$\sum_{i=1}^{m}a_ix_i=\sum_{j=1}^{n}b_iy_i$
for $a_i,b_j\in S,$ such that there exist $x_1,...,x_m,y_1,...,y_m$ such that the formula is not satisfied?
EDIT I have asked a follow-up question here.