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From: Andrei Popescu <uuomul@yahoo.com>
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Subject: projective algebras
Date: Tue, 25 Feb 2003 11:55:01 -0800 (PST)
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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