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 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). 

 

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

 

 

 

 

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