These sound like results of Walter Taylor from the early '70's
publshed in Algebra Universalis


At 12:24 PM +0000 5/15/05, univalg@yahoogroups.com wrote:
>There is 1 message in this issue.
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>Topics in this digest:
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> 1. absolute retracts in varieties of algebras
> From: "p_ouwehand" <peter@...>
>
>
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>Message: 1
> Date: Sun, 15 May 2005 08:27:39 -0000
> From: "p_ouwehand" <peter@...>
>Subject: 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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