- 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 avariety of algebras: An algebra A is an absolute retract if and onlyif it is (i) equationally compact, and (ii) algebraically closed(i.e. every finite set of equations satisfiable in some extension ofA is already satisfiable in A). In particular, a finite algebra is anabsolute retract if and only if it it is algebraically closed.[...](1) Are the above resultsalready known (and where can I find them)?, and (2) Are there anysignificant papers that study algebraically closed algebras ingeneral (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 isLocally 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 inHodges, 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 HebertLet me know if a more precisely written proof is needed
**From**univalg@yahoogroups.com **To**univalg@yahoogroups.com **Cc****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)?>>>>>> Yahoo! Groups Links>> <*> To visit your group on the web, go to:> http://groups.yahoo.com/group/univalg/>> <*> To unsubscribe from this group, send an email to:> univalg-unsubscribe@yahoogroups.com>> <*> Your use of Yahoo! Groups is subject to:> http://docs.yahoo.com/info/terms/>>>