From: Andrei Popescu <uuomul@yahoo.com>
To: categories@mta.ca
Subject: projective algebras
Date: Tue, 25 Feb 2003 11:55:01 -0800 (PST) [thread overview]
Message-ID: <20030225195501.74658.qmail@web12001.mail.yahoo.com> (raw)
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
next reply other threads:[~2003-02-25 19:55 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
2003-02-25 19:55 Andrei Popescu [this message]
2003-02-26 10:24 ` Mamuka Jibladze
2003-03-03 9:43 ` Inductive datatypes in toposes Lutz Schroeder
2003-03-03 16:23 ` Luigi Santocanale
2003-03-03 17:30 ` Thomas Streicher
2003-03-04 1:49 ` Alex Simpson
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