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

WALTER TAYLOR 37
MODEL-THEORETIC QUESTIONS
Sections 5-13 have emphasized properties of equations per se (although of course,
every property of [a set of] equations is a property of varieties, and vice versa). We
now turn to some investigations which have emphasized the models of the equations.
Here the lines are less sharp. Completely general descriptions of models in equationally
defined classes do not exist - but on the other hand the very complete existing
investigations of models of special equational theories (e.g. groups, lattices, Boolean
algebras) must be omitted from this survey on the ground of space alone! For studies
in the (special) model theory of various less well known varieties, one need only look
in almost any issue of Algebra Universalis. Most of the results we will report on will be
of medium generality, i.e. valid for an interesting class of varieties, but not for all
varieties. It is also hard to say where the model theory of equational classes stops and
general model theory begins.
14. Some further invariants of the equivalence class of a variety. See õ 7 above
for equivalence; with the exception of õ õ 5, 8, 10, 11, 12, we have been writing of the
equivalence class of a variety V. In this section we survey very quickly some other
equivalence invariants which have been studied.
14.1. The spectrum of V: spec V= (nco' (there exists AV)lAl=n).
Clearly 1 6 spec V, and since V is closed under the formation of products, spec V is
multiplicatively closed. Gr/itzer [161] proved that, conversely, any multiplicatively
closed set containing 1 is the spectrum of some variety (see also [ 119 ] ). Froemke and
Quackenbush [ 144] showed that this variety need have only one binary operation.
The characterization of sets spec V for V j'nitely based seems to be much more
difficult. McKenzie proved [296] that if K C_ co is the spectrum of any first order
sentence, then there exists a single identity o = r such that the multiplicative closure
of K tJ ( 1 ) is the spectrum of (o = r). (See also [328] .) Characterizations of first
order spectra are known in terms of time-bounded machine recognizability - see [ 130]
for detailed statements and further references. Note that the definition of spec can be
extended to mean the image of any forgetful functor (or any pseudo-elementary class)
- see [ 130] [ 131 ] for more details. For example, we can consider
T(V) = (A: A is a topological space and there exists
(A,F t) 6 V with all F t continuous).