Re: projective algebras

["Mamuka Jibladze" <jib@rmi.acnet.ge>]

Here is some more information on the question of Andrei Popescu. In the
paper "Projectives are free for nilpotent algebraic theories" by T.
Pirashvili (pages 589-599 in: Algebraic $K$-theory and its applications.
Proceedings of the workshop and symposium, ICTP, Trieste, Italy, September
1-19, 1997. World Scientific, Singapore (1999)) it is shown that if
projectives are free in the category of internal abelian groups of a
sufficiently nice (e. g. Maltsev) variety, then the same holds for the
category of all nilpotent (in the sense of commutator calculus) algebras in
that variety. This is derived from the fact that one can lift splittings of
idempotents along a linear extension of algebraic theories.

Mamuka

> 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