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

14 EQUATIONAL LOGIC
groups with a specific presentation abstract groups
is closely analogous to the passage
varieties, as defined by a set of laws algebraic theories.
Although this analogy is perfect for operation symbols and laws, unfortunately the
models (i.e., the algebras in a given variety) fit more conveniently with the L.H.S., and
somewhat spoil the analogy (although they correspond, very roughly, to the structure
which is to be preserved in forming a group of automorphisms). Moreover
equivalence-invariance of model theoretic properties is almost always transparent; and
such properties can usually be discussed in a very simple invariant way (by considering
the set of all operations - see e.g. [282] ); cf. õ õ 14-16 below.
8. Bases and generic algebras. As seen in õ5, if 230 is any set of sentences, the
smallest equational theory _ Z0 is
Eq Mod 0 = {e: 0 [- e},
and in this case we say that 0 is a set ofaxiorns, or an equational base for . Several
of the next sections are concerned with the problem of finding (various sorts of) bases
Z0'
Here we consider what amounts to some concrete examples of Birkhoff's
theorem of õ 3, namely we look for a base 0 for a single algebra A, i.e., we want
Mod 0 = HSP A.
Actually, given 230, A may be regarded as unknown. Here we refer to A as genteric for
the variety Mod 0 or for the theory
every variety V has a generic algebra, i.e.
= Eq Mod 0' Using P one can easily see that
for all V there exists A(V = HSP A).
One such A is the V-free algebra on q0 generators; see also [408]. We mention here a
few examples of such A and 0'
8.1. The ring Z of integers is generic for the theory of commutative rings.
8.2. The 2-element Boolean ring [with unit] is generic for the theory of Boolean
rings [with unit], given by the laws for rings [with unit] together with the law x 2 = x.
(Similarly for Boolean algebras, by remarks in 87; this fact may be interpre'ted as a
completeness theorem for propositional logic.)