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

38 EQUATIONAL LOGIC
(Some preliminary investigations on T(V) appear in [429] ;cf. õ 16 below.) Of course,
many descriptions of individual varieties in the literature yield spec (V). A certain
amount of attention has. focused on the condition spec (V) = {1}. (See e.g. [231 ], [9]
and remarks and references given in [420, page 382]; cf. 12.2 above.) For instance,
Austin's equation [ 9 ]
((y2.y). x)((y2.(y2.y)). z) = x
has infinite models but no nontrivial finite models.
Mendelsohn [310] has shown that if V is an idempotent binary variety given by
2-variable equations, then spec V is ultimately periodic.
14.2. The fine spectrum of V is the function
fv(n) = the number of non-isomorphic algebras of power n in V.
Characterization of such functions seems hopeless. A typical theorem is that of
Fajtlowicz [133] (see also [424, pages 299-300] for a proof): if fv(n) = 1 for all
cardinals n > 1, then V must be (within equivalence) one of two varieties: "sets" (no
operations at all) or "pointed sets" (one unary operation f which obeys the law
fx = fy). For some related results see Taylor [424], Quackenbush [371], McKenzie
[300] and Clark and Krauss [90].
PROBLEM. [424]. Does the collection of all fine spectra form a closed subset
of co co (power of a discrete space)?
14.3. Categoricity in power. Varieties obeying the condition fv(n)= 1 for all
infinite n > the cardinality of the similarity type of V have been characterized (within
equivalence) by Givant [156] and Palyutin [345]. For a detailed statement, also see
e.g. [424, page 299]. For example if V is defined by the laws (of Evans [ 118] )
d(x,x) = x
d(d(x,y),d(u,v)) = d(x,v)
(*) c(c(x)) = x
c(d(x,y)) = d(cy,cx),
then every algebra in V is isomorphic to an algebras with "square" universe A X A on
which
c((c,fi)) = (fi,)
d((a,/3),(7,6)) = (a,6).