Cartesian map factorization in Set extensions. Applications in computer science and efficient language construction. J. E. P ALOMAR T ARANC ´ ON Abstract: In the scope of many-valued logic, we show some results giving rise to Cartesian factorization algorithms for maps and binary relations. The obtained methods work handling symbols blindly; therefore the underlying algebraic struc- tures can be disregarded. Cartesian factorizations matter, due to their applications in optimizing those algebraic structures used in computer science. The advantages of Cartesian factorizations in algorithm construction can be compared with the advantages using the binary numeral system with the Roman one. In addition, by means of Cartesian factorizations the size of databases and storage devices can be strongly reduced. 2000 Mathematics Subject Classification 18B10 (primary); 68R05, 3B50 (sec- ondary) Keywords: Morphism factorization, many-valued logic, efficient languages, com- puter science 1 Introduction The main aim of this article consists of investigating adequate results to factorize maps into Cartesian products, in the scope of many–valued logic. Sometimes, these factor- izations can be carried out knowing some properties of the domain members. In [4] this topic is investigated through homomorphisms and abstractions. A particular case of this procedure consists of factorizing those maps defined by a polynomial p(x), in a finite subset of N , into a Cartesian product f 1 × f 2 ×··· f n . This factorization is possible, when each f k is the homomorphic image of p(x) in the field Z k of integers modulo k , provided that k is prime. Unfortunately, the map factorization requires handling the algebraic structure of the involved maps, and for big domains, the polyno- mial p(x) can be too large. Theorem 3.5 shows sufficient conditions to build efficient Cartesian–factorization algorithms disregarding the associated polynomial p(x). Only adequate domain partitions matter. Nevertheless, new constructing strategies should be investigated, to improve the factorization algorithms based upon Theorem 3.5 because of their applications in Computer Science. This claim needs some explanation.