The existence of an algebra whose set of identities and first order theory are equivalent Is there an algebra $A$ (for example a group) such that $Th(A)$ is logically equivalent to $id(A)$? In other words, is there an algebra $A$ such that
$$
Mod(Th(A))=Var(A)?
$$
Clearly finite algebras do not have this property. It seems that such an algebra should be relatively free.
This question is related to my previous two questions 


*

*Relatively free algebras in a variety generated by a single algebra

*relatively free groups in $Var(S_3)$
Edition: Only trivial algebra has this property by the comment of Gerhard Paseman. So I ask again the question by $\pm id(A)$ instead of $id(A)$. Is there any algebra A (especially a group) such that $Th(A)$ is logically equivalent to $\pm id(A)$? Here $\pm id(A)$ means the set of identities and negated identities. 
P.S. By negated identity  I mean a sentence 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. Is there any negated identity in a non-trivial algebra? Clearly there are no negated identities in groups but if we add constants to the language of groups there will be many negated identities.
 A: I imagine that definitions of $Mod, Th, Var$ and so on have not changed since I saw them decades ago.  The trivial one element algebra in any finite type (and likely any infinite type) is an easy example which satisfies $Mod(Th(\textbf{A})) \approx Var(\textbf{A})$.  Since it is expected that $Th(\textbf{A})$ is strong enough to indicate whether $\textbf{A}$ has more than one element, this is the only example to be expected (Thank you Joel).
Gerhard "Ask Me About Trivial Algebra" Paseman, 2014.01.09
A: For an example with a negated identity, let $A$ be a vector space over an infinite field in the usual signature ($+$ and scalar multiplications) together with an additional constant $1\ne0$.
EDIT: Since it was apparently not obvious (judging from the comment), this is meant to be an example of an algebra whose full first-order theory is equivalent to its set of valid identities and negated identities. (This follows here from the fact that the theory of infinite vector spaces over a field $F$ is categorical in every cardinality $\kappa>|F|$.)
A: Here is another example. Let $A$ be the algebra consisting of infinitely many distinct constants $c_n$. The theory of this model is logically equivalent to the assertions $c_n\neq c_m$ for distinct $n,m$, and so the models of $\text{Th}(A)$ are the same as the models of $\pm\text{id}(A)$.
