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?

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    $\begingroup$ A variety in a signature without constants will contain the empty algebra. But the empty set is not a group. $\endgroup$ – Zhen Lin Feb 15 '16 at 2:39
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    $\begingroup$ @ZhenLin Not necessarily. Some presentations forbid empty domains. $\endgroup$ – Pedro Sánchez Terraf Feb 15 '16 at 4:54
  • $\begingroup$ Just a side remark. Finite groups do form a variety of finite monoids (also called pseudovariety of monoids), that is, a class of finite monoids closed under homomorphisms, submonoids and finite direct products. $\endgroup$ – J.-E. Pin Feb 16 '16 at 7:13

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.

  • $\begingroup$ @EricWofsey Thanks for the edit! $\endgroup$ – Pedro Sánchez Terraf Feb 15 '16 at 13:53

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