We elucidate a close connection between the Theory of Judgment Aggregation and a relatively young but rapidly growing field of universal algebra, that was primarily developed to investigate constraint satisfaction problems. We show that theorems in the above field translate (often directly) to impossibility, classification and robustness theorems in social choice theory. We refine the classification of E. Dokow, R. Holzman of binary evaluations, complete their classification theorem for non-binary evaluations, give a new classification theorem for the majoritarian aggregator and show how Sen's well known theorem follows from it, define new aggregator classes and also prove theorems about them. We also give upper bounds on the complexity of computing if a domain is impossible or not.