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
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.