  We elucidate a close connection between the Theory of Judgment Aggregation
(more generally, Evaluation Aggregation), and a relatively young but rapidly
growing field of universal algebra, that was primarily developed to investigate
constraint satisfaction problems. Our connection yields a full classification
of non-binary evaluations into possibility and impossibility domains both under
the idempotent and the supportive conditions. Prior to the current result E.
Dokow and R. Holzman nearly classified non-binary evaluations in the supportive
case, by combinatorial means. The algebraic approach gives us new insights to
the easier binary case as well, which had been fully classified by the above
authors. Our algebraic view lets us put forth a suggestion about a
strengthening of the Non-dictatorship criterion, that helps us avoid "outliers"
like the affine subspace. Finally, we give upper bounds on the complexity of
computing if a domain is impossible or not (to our best knowledge no finite
time bounds were given earlier).
