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 

1. http://mathoverflow.net/questions/153743/relatively-free-algebras-in-a-variety-generated-by-a-single-algebra

2. http://mathoverflow.net/questions/153883/relatively-free-groups-in-vars-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.