Universal Algebra/Definitions, examples

Recall that whenever $$S$$ is a set, then $$\mathcal T(S)$$ is the class of all $$S$$-tuples, regardless of size.

Note that in this definition, $$\alpha = \{\}$$ (the set consisting only of the empty tuple) is allowed, so that in this case, $$\circ$$ can be identified with a constant in $$A$$. It is customary to regard $$0$$-ary operations as constants in $$A$$.

For example, if we have a subset of a group that contains the identity and is closed under inversion and the product (that is, if we have a subset of a group that is closed under the 0-ary, the 1-ary and the 2-ary operation), then that subset is a subgroup.