Subject: Gratzer

From: Szendrei Agnes <[email-masked]>
Date: Mon, 30 Jul 2007 16:46:42 +0200 (CEST)

Hi Dear,

I posted the pdf file of Gratzer's Mal'tsev-type paper
on my SZEGED web page at

http://www.math.u-szeged.hu/~szendrei/gratzer.pdf

(I was not able to sftp the file over to spot.
I don't understand why, but on spot I kept getting a pdf
file of lenght 0.
I tried to sftp directly from szeged, and also through
euclid -- none of them worked.)

--------------------------

I checked

Author: Gratzer
Anywhere: Malcev or Maltsev

on MathSciNet.

I found only one paper by Gratzer where the results
is a Maltsev-type condition, the one we already know about.
In all other papers that were returned for these searches
either Maltsev was mentioned in a different context
(like Maltsev's lemma on congruence generation
or Maltsev product of varieties) or a Maltsev-type
condition of someone else (e.g. Day) was referred to.

---------------------------

I copy here M. Gould's review on Gratzer's Maltsev-type
paper, which seems to confirm that it was Gratzer who first
gave a formal definition for a Maltsev-type condition.

Motivated by the form of a classical theorem of
Mal\cprime cev and of subsequent results by Jónsson
and others, the author frames the following definition:
"A Mal\cprime cev-type condition of equational classes
is of the form `$(P_n)$ there exists a natural number $n$,
and polynomial symbols $p_0,\cdots,p_{m_n-1}$ satisfying
a set of identities $\Sigma_n$', where $(P_{n+1})$ is
weaker than $(P_n)$ and the form of $\Sigma_n$ is independent
of the type of algebras considered."

Following Tarski, he then defines regularity of an equational
class to mean that for every pair of congruences on an algebra
in the class, if the pair have a congruence class in common,
then they are equal. Weak regularity is defined to mean that
every algebra in the class contains an element having the property
that any two congruence relations having the same class containing
that element are equal. The paper's main results provide formulations
of these two conditions as Mal\cprime cev-type conditions.

Reviewed by M. Gould
-------------------------------------

Agi