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

42 EQUATIONAL LOGIC
a congruence on A such that 0 = (3 B 2. Abelian groups and distributive lattices
have CEP, but groups and lattices do not. See e.g. Banaschewski [30], Pigozzi [355],
Davey [ 97 ], Day [ 101 ], [ 104], Fried, Griitzer and Quackenbush [ 140], Bacsich and
Rowlands-Hughes [15], Magari [275], Mazzanti [289] (where one will find some
other references to the Italian school - in Italian usage, "regolare" means "having the
CEP"). In [ 15 ] there is a syntactic characterization of CEP in the style of that for AP
in 14.6 above, although these two properties are really rather different. This
characterization is closely related to Day [ 101]. Notice that in Boolean algebras the
CEP can be checked rather easily because every algebra B is a subalgebra of some
power A I, where A is the two-element algebra and every congruence 0 on B is of the
form
(ai)0(b i) iff { i' a i = b i} G F
for some filter F of subsets of I. (And, of course, the same filter F may be used to
extend 0 to larger algebras.) A variety in which congruences can be described by filters
in this manner is calledfiltral - e.g. [140], [289], [275] and especially [39]. But, e.g.,
semilattices form a non-filtral variety which has CEP. It is open whether filtrality
implies congruence-distributivity ( õ 15 below). For CEP see also Stralka [405 ].
PROBLEM. (Griitzer [ 165, page 192] ). If V satisfies
(for all X C_ V) HS X = SH X,
then does V have the CEP? This is true for lattice varieties (Wille).
14.8. A variety V is residually small iff V contains only a set of subdirectly
irreducible (s.i.) algebras (õ4), i.e. the s.i. algebras do not form a proper class, i.e.
there is a bound on their cardinality. It turns out that this bound, if it exists, may be
taken as 2 n, where n = N 0 + the number of operations in V (see [419] ). V is residually
small iff V 0 can be taken to be aset in (*) of õ4, which is to say that Vhas a "good"
coordinate representation system. For some other conditions equivalent to residual
smallness, see [419] and [34]; also see [24] where e.g. finite bounds on s.i. algebras
are considered. McKenzie and Shelah [301] consider bounds on the size of simple
algebras in V and obtain a result analogous to that on 2 n just above. (An algebra is
simple iff it is non-trivial and has no proper homomorphic image other than a trivial
algebra. Every non-trivial variety has at least one simple algebra [273] ;but see [326]