Is groups with binary operation alone a variety?

Question

In the signature (+, 0, -), the class of groups are a variety, because they can be defined by a set of universal equations. But is it already a variety in the signature (+), by itself? The more important question is, is there a standard terminology in the universal algebra literature for a class of functions and/or constants that are not necessarily a variety by themselves, but are so when augmented by further functions and/or constants?

Answer 1

It won't be a variety. As you observe, with the signature $(+,0,-)$ (or even without the constant) you can axiomatize the class of groups using universal sentences, but with $+$ only it is not possible: $\mathbb{Z}$ is a group, but its substructure $\mathbb{N}$ is not. And universal sentences are preserved by taking substructures.

Concerning terminology, the general concept of reduct is applicable. So we say that the class of all groups in the signature $+$ is a reduct of a variety.