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

WALTER TAYLOR 29
12.12. Mod ; is residually small (cf. 14.8 below) (McNulty [306] ).
PROBLEM 4. (Mycielski - see [197]). Is it decidable whether a finite set of
terms is jointly c-universal 0c a fixed cardinal)?
(Terms ri(x 1 ,...,Xni) (1 i n) in operations F 1 ,...,F s are jointly c-universal iff
for any operations Gi: c ni - c, the operations F 1 ,...,F s can be defined on c so that the
term r i defines G i (1 i n). This notion of universality has been very useful in the
study of undecidability of properties of sets of equations - see McNulty [305] .)
Finally, we mention that Burris and Sankappanavar [80] have investigated
undecidability properties of congruence lattices and lattices of subvarieties (õ 13 just
below). A sample result: in a similarity type with at least one operation of rank 2,
the lattice A of all equational theories has a hereditarily undecidable first order
theory.
13. The lattice of equational theories. For a fixed type (nt)tCT, order the
family A of all equational theories by inclusion. One easily sees (from Birkhoff's
Theorem of õ5 or directly from the definition at the beginning of õ5), that A is
closed under arbitrary intersections - and so from purely lattice theoretic
considerations, A has arbitrary joins as well, and so is a complete lattice. More
specifically,
iGVlEi = {e' iGoi;i [- e} = Eq(iiMod Ei).
From the proof-theoretic characterization of v it follows that A is an algebraic closure
system and hence an algebraic lattice (see [91] [163] or [165]). Specifically, the
compact elements of A are the finitely based theories of õ9 above, and every element
is the join (actually the union) of all its finitely based subtheories. Obviously the join
of two finitely based theories is finitely based (this holds for compact elements in any
lattice); but the meet (i.e., intersection) of two finitely based theories can fail to be
finitely based. We present an example of Karnofsky (unpublished - see [354]) (here
and below we will sometimes express a theory by one of its finite bases without
further mention)' Z 1 ß x(yz) = (xy)z ;2' x(yz) = (xy)z
(xyz) 2 = x2y2z2 x3y 3 = y3x3
x3y3z2w 3 = y3x3z2w 3.