During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have in my mind:


Let $\mathcal{L}$ be an algebraic language. A negated identity in $\mathcal{L}$ is a formula of the form
$$
\forall x\_1 \ldots \forall x\_n: p(x\_1, \ldots, x\_n)\neq q(x\_1, \ldots, x\_n),
$$
where $p$ and $q$ are terms in $\mathcal{L}$. For an algebra $A$ of type $\mathcal{L}$ assume that $id^+(A)$ and $id^-(A)$ are the set of all identities and negated identities valied in $A$, respectively. 
As we know 
$$
Var(A)=Mod(id^+(A))
$$
is the variety generated by $A$ and during the questions


1. [Relatively free algebras in a variety generated by a single algebra](https://mathoverflow.net/questions/153743/relatively-free-algebras-in-a-variety-generated-by-a-single-algebra)
2. [relatively free groups in $Var(S\_3)$](https://mathoverflow.net/questions/153883/relatively-free-groups-in-vars-3)


we tried to understand the relative free algebras in $Var(A)$. Now, suppose 
$$
Var^-(A)=Mod(id^-(A)).
$$
It is easy to see that $Var^-(A)$ is $\{ S, P\}$-closed and so it is a pre-variety. Hence, for any set $X$, there exists a free algebra $F\_{Var^-(A)}(X)$ in this pre-variety. Now, we ask: 


1. How we can determine the structure of $F\_{Var^-(A)}(X)$? I mean an answer like polynomial functions given by Anton Klaychko or the extendblity criterion given by Benjamin Steinberg in my previous question.
2. Can we characterize $Var^-(A)$ using class operators? I mean some thing like HSP.


P.S. To answer the question $F\_{Var^-(A)}(X)=$?, one should determine the set of all identities with variables from $X$ which are logical consequence of the set $id^-(A)$. More precisely, suppose 
$$
R(X)=\{ (p, q):\ id^-(A)\vDash p=q\}.
$$
Then we have $F\_{Var^-(A)}(X)=T\_{\mathcal{L}}(X)/R(X)$, where $T\_{\mathcal{L}}(X)$ is the term algebra. So, to answer the first question, one should say that: which identities are logical consequences of given negated identities. For example, suppose $\mathcal{L}=(0,1, +, -, \times)$. Is it possible to determine all non-trivial identities which can be deduce from the negated identity
$$
\forall x \forall y: x^2+y^2\neq-1?
$$
Honestly, I have no even one example of non-trivial identity deducible from the above negated identity.



Negated varieties and their relatively free algebras
This refers to the PS in the question (I cannot make comments): the term algebra $T\_L(X)$ seems to satisfy the sentence $\forall x\forall y: x^2+y^2\ne -1$ but $T\_L(X)$ does not satisfy any non-trivial identity. It follows that no non-trivial identity is a consequence of that sentence.


