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

12 EQUATIONAL LOGIC
algebras do not matter, and some writers save themselves this and related
considerations by always taking algebras to be non-empty. See Friedman [ 142] for a
more general theory of definition within varieties. Operations may be implicitly
definable (i.e., specified by the other operations) in V, but not explicitly definable
except by arbitrarily complex formulas of first order logic.)
Equivalence has its model-theoretic aspect, too. Varieties V 1 and V 2 are
equivalent (i.e. Eq V 1 and Eq V 2 are equivalent in the above sense) iff there exists an
isomorphism of categories : V 1 - V 2 which commutes with the forgetful functor to
sets (i.e. V 1 has the same universe as V 1 and a similar fact holds for
homomorphisms). (These categories are formed from the non-empty models of a
variety and all the homomorphisms between them. Cf. the remarks in the 2nd
paragraph on page 52 of [425]. For various references and remarks on this theorem of
A. I. Malcev, see [420, page 355].) Perhaps the first historical example of an
equivalence of varieties is the well known natural correspondence between Boolean
algebras and Boolean rings (with unit). Also consider the correspondence between the
varieties of Abelian groups and Z-modules - here the equivalence is so easy that some
people write as if it were an equality. Some other interesting examples of equivalence
may be found in [96].
Very close to the idea of equivalence (in it model theoretic form) is the idea of
weak isomorphism as developed in Wroctaw. This together with an emphasis on
independent sets over free algebras gave equational logic a somewhat different
direction and flavor in that school. See Marczewski [282] for an introduction to these
ideas. Briefly, algebras A and B are weakly isomorphic iff there is a bijection 0: A - B
such that the algebraic operations of A are exactly the same as the operations
0-1F(0Xl .... ,0Xn) where F(Xl,...,Xn) is an algebraic operation of B. (Here, by the
family of algebraic operations, we mean the closure under composition of the family
of all operations F t together With all projection functions.) Then two varieties are
equivalent iff they have weakly isomorphic generic algebras (see õ 8 for "generic").
Properties of varieties seem more natural and interesting if they are
equivalence-invariant, if only because then they do not force us to make any
"unnatural" choice between, say 11 and 12 above. For example, the similarity type