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## absolute retracts in varieties of algebras

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• I have a simple characterization of the absolute retracts in a variety of algebras: An algebra A is an absolute retract if and only if it is (i) equationally
Message 1 of 2 , May 15, 2005
I have a simple characterization of the absolute retracts in a
variety of algebras: An algebra A is an absolute retract if and only
if it is (i) equationally compact, and (ii) algebraically closed
(i.e. every finite set of equations satisfiable in some extension of
A is already satisfiable in A). In particular, a finite algebra is an
absolute retract if and only if it it is algebraically closed.

From previous work, I know that in a congruence distributive variety
every finite absolute retract is a product of maximal subdirectly
irreducibles (i.e. those s.i.'s that have no proper essential
extensions), and that in lattice varieties, the converse holds as
well.

Thus in a variety of lattices, the finite algebraically closed
lattices are precisely products of maximal subdirectly irreducibles.
E.g., the finite AC distributive lattices are exactly the finite
Boolean lattices, a result apparently due to Schmid (1979).

What I would like to know is the following: (1) Are the above results
already known (and where can I find them)?, and (2) Are there any
significant papers that study algebraically closed algebras in
general (particularly the congruence distributive case)?
• p_ouwehand asked: I have a simple characterization of the absolute retracts in a variety of algebras: An algebra A is an absolute retract if and only if
Message 2 of 2 , May 15, 2005

" I have a simple characterization of the absolute retracts in a
variety of algebras: An algebra A is an absolute retract if and only
if it is (i) equationally compact, and (ii) algebraically closed
(i.e. every finite set of equations satisfiable in some extension of
A is already satisfiable in A). In particular, a finite algebra is an
absolute retract if and only if it it is algebraically closed.
[...]
(1) Are the above results
already known (and where can I find them)?, and (2) Are there any
significant papers that study algebraically closed algebras in
general (particularly the congruence distributive case)? "

This characterization actually holds in the wider context of accessible categories with pushouts. (Those are more general than  locally presentable categories , which in turn are something a little bit more general than quasivarieties.)

One of the problems here is the abundance of terminology for very close (or even equivalent) concepts.
In particular an algebraically closed embeddings are also called a pure embeddings, and this has been studied quite a lot in locally presentable categories. An excellent general reference is

Locally presentable and accessible categories, by J.Adamek and J. Rosicky, London MAth Soc Lecture Note Series 189, 1994.

More specific works on your question would be my
(1) Algebraically closed and existentially closed substructures in categorical context, Theory and Applications of Categories 12 (2004), 269-298 ( pdf )
and
(2) K-purity and orthogonality, Theory and Applications of Categories 12 (2004), 355-371 (pdf)

In case you are more familiar with the classical model-theoretic notations, a proof that "equationally compact" is equivalent to "every pure embedding is a retract" in a wide context is in
Hodges, Model Theory1993 p. 528.
That this is equivalent to "pure-injective" follows from the fact that pushouts preserve pure embeddings (and maybe by model-theoretic methods too).

Your result may not be mentioned explicitely in any of the above,  but it now reads as
"Algebraically closed + pure-injectivity  <=>  Absolute retracts"
which follows immediately from some of the  known equivalent definitions of purity, recalled in (1) and (2).

Michel Hebert

Let me know if a more precisely written proof is needed

 From univalg@yahoogroups.com
 To univalg@yahoogroups.com
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 Date Sun, 15 May 2005 08:27:39 -0000
 Subject [univalg] absolute retracts in varieties of algebras
> I have a simple characterization of the absolute retracts in a
> variety of algebras: An algebra A is an absolute retract if and only
> if it is (i) equationally compact, and (ii) algebraically closed
> (i.e. every finite set of equations satisfiable in some extension of
> A is already satisfiable in A). In particular, a finite algebra is an
> absolute retract if and only if it it is algebraically closed.
>
> From previous work, I know that in a congruence distributive variety
> every finite absolute retract is a product of maximal subdirectly
> irreducibles (i.e. those s.i.'s that have no proper essential
> extensions), and that in lattice varieties, the converse holds as
> well.
>
> Thus in a variety of lattices, the finite algebraically closed
> lattices are precisely products of maximal subdirectly irreducibles.
> E.g., the finite AC distributive lattices are exactly the finite
> Boolean lattices, a result apparently due to Schmid (1979).
>
> What I would like to know is the following: (1) Are the above results
> already known (and where can I find them)?, and (2) Are there any
> significant papers that study algebraically closed algebras in
> general (particularly the congruence distributive case)?
>
>
>
>
>
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