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

WALTER TAYLOR 45
has an influence on the fine spectrum (cf. 14.2) - see [424, pages 285-6]. For the
closely related subject of "cancellation," see Lova'sz [262].
14.11. Universal varieties. We must refer the reader to the papers [359], [361]
of Pigozzi for this relatively new notion which promises to be quite important. F is
universal iff for every similarity type there exist F-terms o t corresponding to the
operations of this type such that for each A of this type there exists B C F such that A
is a subalgebra of (B; FBt)tC T and (B; FBt)t T obeys exactly the same laws as A. (E.g.,
the variety of quasigroups is universal.) Many of the undecidability and
lattice-theoretic results of õ õ 12,13 above extend to universal varieties.
14.12. The Schreier property (all subalgebras of free algebras are free) is
investigated in Meskin [311], Kelenson [243], Aust [8], Jeek [210] and Budkin
[71]; cf. 12.8 above. Neumann and Wiegold (supplementary bibliography) showed
that the only Schreier varieties of groups are all groups, all Abelian groups, and all
Abelian groups of exponent p (prime). (Schreier earlier proved that the variety of all
groups has this property.) T. Evans gave a parallel result for semigroups (see
supplementary bibliography).
15. Malcev conditions and congruence identities. Malcev proved [278] that a
variety F has permutable congruences iff there is a ternary term p(x,y,z) such that
F p(x,x,y) = p(y,x,x) = y.
(For binary relations , k, define - k = (a,c): there exists b(a,b) and (b,c) k
and say that F has permutable congruences iff - b = ' for all congruences on any
A F.) B. J6nsson proved [221 ] that all congruence lattices of algebras in F obey the
distributive law iff there exist ternary terms Pi(x,y,z) (0 in) such that the
following equations hold identically in F:
Pi(x,y,x) = x (0 i n)
P0(x,y,z) = x pn!X,y,z) = z
Pi(X,X,y) = Pi+ 1 (x,x,y) (i even)
Pi(x,y,y) = Pi+l(x,y,y) (i odd).
(But cf. 12.10 above.) And Day [98] proved a similar result for modularity of the
congruence lattice. ,Properties of varieties definable in this way by the existence of
terms have come to be known as Malcev-definable (see [420], [333] or [26] for a