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

24 EQUATIONAL LOGIC
is an interval of natural numbers. All these cases can occur. Referring to 9.25 above
(from [413] ),
V(To) = { RO }
V(T 1) = 0.
For some other infinite irredundant bases, see [ 1 1 ], [78 ], [ 120], [ 198 ], and [349 ].
Distinct subsets of an irredundant base define distinct subtheories of 23, and so infinite
irredundant bases are useful in proving that some lattices of varieties have cardinality
2N0;see 13.3, 13.4 and 13.11 below.
McKenzie proved that V(23) can be any interval, and Ng showed that 23 can be
found with one binary operation (see [413] ). For example, if
23 = œF5Xl.--x6 = F5x2---X6Xl} ,
then V(23) = { 1,2}, essentially because the cyclic group C 6 has both a single generator
and an irredundant set of two generators. V(23) is an unbounded interval if 23 I- r = x,
where r contains x at least twice (Tarski [413]), strengthened by McNulty [305] to
the case where r has at least one operation of rank > 2. On the other hand, if 23 is
defined by balanced equations, and 23 is finRely based, then V(23) is a bounded
interval [ 305 ]. (An equation o = r is balanced iff each variable, each nullary operation
symbol and each unary operation symbol occurs equally often in o and r.) T. C. Green
got irredundant bases of power n (any n C co) for groups ([172], see also [413]).
V(23) was also defined by G. Grfitzer [163, Problem 34], who asked for a
characterization of it. Finally, note that V is not an equivalence invariant (õ 7), as can
be seen from 10.1 and 10.8 or from 10.3 and 10.10.
1 2. Decidability question. We assume given a finite alphabet A and a fixed way
of interpreting all our variables, function symbols, terms, equations, etc.
unambiguously as words in A. We assume that the reader knows what is meant for a
collection W 0 of words to be decidable (relative to a collection W D_ W 0 of words). For
this (non-numerical) notion of decidability, probably the Turing-machine approach
("computability") is easiest. For readable brief descriptions, see [43 1 ], [2 1 7], [267],
or [416, pages 12-14]. Usually W will be obvious (such as the collection of all finite
sequences of equations) and we will not mention it. As is well known [op. cit. ] a
relatively easy analysis of Cantor's diagonal argument yields undecidable sets; the