An "holoid" is a commutative monoid in which division is a partial order. Dubreil, Fuchs, Mitsch and Bosbach studied certain holoids in which every element has a unique factorization (possibly reduced) into irreducible, prime or maxiaml elements. We give a specific meaning to the words "reduction" and "reduced". Then we study a new family of holoids, called factorial -- a concept which generalizes the previous holoids with "unique factorization" --. The most meaningful difference is that we don't suppose any chain condition. However, we have again the "good" properties of these holoids: existence of l.c.m., existence of a minimum solution to the equation ax = b when a divides b. We also prove the following result: if H is factorial, then it is also factorial with respect of l.c.m. as a law of composition.