Survey 1979: Equational Logic - Department of Mathematics ...

WALTER TAYLOR 31
all types by Burris [74] and Je2fek [207], answering Problem 33 of Grtzer [163]).
Second, Bolbot [59] and Jeek [207] proved that (given at least two unary
operations or one operation of rank > 2) A is dually pseudo-atomic, i.e. the zero of A
(i.e. 230 = 0) is the meet of all dual atoms (i.e. equationally complete theories). But see
some of the examples below for varying numbers of equationally complete theories in
various A(Z).
Kalicki and Scott [234] found all equationally complete semigroups; there are
only 0 of them. McNulty used their description in reproving Perkins' result that it is
decidable whether
23, x(yz) = (xy)x [- x = y.
All equationally complete rings were found by Tarski [412]; again, there are 0 of
them. See also [343]. We cannot begin to cover all the information presently known
on equational completeness. For further information, consult Gritzer [163, Chapter
4], or Pigozzi [354, Chapter 2]. Here we sample just a few very recent results.
THEOREM. (Pigozzi [360]). There exists an equationally complete variety
which does not have the amalgamation property. (Answering a question of S.
Fajtlowicz.) (See 14.6 below for the amalgamation property.)
THEOREM. (Clark and Krauss [89]). If V is a locally finite
congruence-permutable equationally complete variety, then V has a plain paraprimal
direct Stone generator. (See [89] for the meaning of these terms - roughly speaking,
this means that V is generated in the manner either of Boolean algebras or of primary
AbelJan groups of exponent p.)
Some examples of known or partly known A(23) for IA(2)1% 2:
13.1. For 23 = 0 in a type with just one unary operation F, Jacobs and
Schwabauer [203] gave a complete description of A. For unary operations and
constants, see Jeek [206].
13.2. If 23 = Eq A for a finite algebra A in a finite similarity type, then Scott
showed [390] that A(23) has only finitely many co-atoms (i.e. equationally complete
* ,
varieties). If A generates a congruence-distributive variety, then Jonsson s lemma (see
õ15 below) easily implies that A(23) is finite. If A is quasiprimal (see [369])then
A(23) is a finite distributive lattice with a unique atom (= HSP{B'B _A}), and,