Dear Categorists, Some time ago, I have posed you a question about the characterization of projective algebras in the category of all algebras of a given signature. Since some of you appeard interested in the subject, I allow myself to send you, in a slightly detailed manner, the answer that I have found. Projective algebras coincide with free algebras in the following cases: I. Any class (i.e. complete subcategory) of algebras that is closed to taking subobjects and for which free algebras exist and have a certain property (namely that there are no infinite chains of elements such that each one is obtained by applying an operation to an n-uple that includes the predecesor in the chain). In particular, II. Suppose X is a countably infinite set. Any quasivariety K of algebras for which the kernel of the unique morphism extending X from the term algebra to the algebra freely generated in K by X has finite congruence classes. In particular, III. - The category of all algebras (of a given signature); - The category of [commutative] semigroups; - The category of [commutative] (non-unital and non-anihilating) semirings. Best regards, Andrei