Can you define a propositional variable as the negation of another variable? Can you have p = not q? 
Is there a rule somewhere that says you can't have this? 
I'm asking this because there's a question that asks me to prove that all positive formulae are satisfiable. A positive formula is a propositional formula that doesn't have a negation anywhere in it. 
But if you can define p = not q then a positive formula p & q would be unsatisfiable. 
 A: Take any "standard" mathematical logic textbook, e.g. :
Dirk van Dalen, Logic and Structure (5th ed - 2013), page 7 :

Definition 2.1.1 The language of propositional logic has an alphabet consisting of :
(i) proposition symbols: $p_0, p_1, p_2,$ . . .,
(ii) connectives: ∧,∨,→,¬,↔,⊥,
(iii) auxiliary symbols: ( , ).
[...] The proposition symbols and $\bot$ stand for the indecomposable propositions, which we call atoms, or atomic propositions.
Definition 2.1.2 The set $PROP$ of propositions is the smallest set $X$ with the properties :
(i) $p_i \in X (i \in \mathbb N), \bot \in X$,
(ii) if $ϕ,ψ \in X$, then $(ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ→ψ), (ϕ↔ψ) \in X$,
(iii) if $ϕ \in X$, then $(¬ϕ) \in X$.

With this definition, we have that a single proposition symbol $p_i$ can be a proposition but, translating into your terminology, a "complex" expression, like $(p_i \land p_j)$ or $(\lnot p_i)$ is not a propositional letter (i.e.a propositional symbol) but a formula (i.e.a proposition).
Note. In order to give a positive answer to your problem, we have to modify the above definition excluding $\bot$.
A: No. A propositional variable stands for a proposition that cannot be broken down into smaller pieces.  You could view "$p = \neg q$" as a definition of a compound proposition $p$, or as an assertion relating two propositional variables $p$ and $q$, but not as a definition of a propositional variable $p$.
To determine whether a compound proposition is positive, you need to break it down all the way into propositional variables.  So if $p$ and $q$ are propositional variables, then $p \wedge q$ is positive, but if $p$ is a compound proposition $\neg q$, then $p \wedge q$ is equal to $\neg q \wedge q$, so it is not positive.
It would be less confusing to use different letters (e.g. $p$, $q$, etc. for propositional variables and $\varphi$, $\psi$, etc. for compound propositions,) but I don't think there is a universal standard for this.
A: p&q is logically equivalent to (not q)&q which is not a positive formula. p&q without the condition of q= not p is a positive formula, with that condition it is not a positive formula and thus is not necessarily satisfiable.
