Survey 1979: Equational Logic - Department of Mathematics ...
Survey 1979: Equational Logic - Department of Mathematics ...
Survey 1979: Equational Logic - Department of Mathematics ...
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50 EQUATIONAL LOGIC<br />
17.3. See e.g. [193], [363] and references given there for "local" varieties.<br />
17.4. There is a rapidly growing theory <strong>of</strong> combinatorial designs as varieties.<br />
Bruck first observed that if a binary operation obeys the laws<br />
then one has a Steiner triple system<br />
XX----X<br />
xy = yx<br />
x(xy) = y<br />
({x,y,xy }: x 4:y},<br />
and conversely, all Steiner triple systems arise in this way. This idea can be greatly<br />
extended; consult Evans [126], Ganter and Werner [148], [149], Quackenbush<br />
[370], Banaschewski and Nelson [35], and Ganter [147].<br />
17.5. Goodstein [159] propounded an equational axiom system for the natural<br />
numbers, equally as strong as Peano's. It differs from the systems described here in<br />
that it had another rule <strong>of</strong> pro<strong>of</strong>, corresponding to uniqueness for inductive<br />
definitions.<br />
17.6. An equation calculus for the recursive definition <strong>of</strong> functions was<br />
developed by G6del and Herbrand -it is described in [246, õ54]. But it differs widely<br />
from the logic <strong>of</strong> equations described here - for example, it lacks the symmetric law -<br />
an equation f(.-.) = g(---) is an instruction meaning roughly, "if the RHS has been<br />
calculated, then regard this as a way <strong>of</strong> calculating the LHS," and this process cannot<br />
be reversed. Of course, this calculus greatly resembles the use <strong>of</strong> equations in e.g.<br />
Fortran, although there are obvious differences.<br />
17.7. Henkin has proposed a theory <strong>of</strong> types which can be viewed as an<br />
equational theory. (See [ 182] .)<br />
17.8. A theory <strong>of</strong> "heterogeneous" varieties (objects <strong>of</strong> more than one type) was<br />
developed by G. Birkh<strong>of</strong>f and J. D. Lipson, A. I. Malcev, and P. J. Higgins. It was<br />
useful in Taylor [420] (q.v. for full references, pages 358-359). Boardman [58] used<br />
essentially the same theory in algebraic topology (his "colors").<br />
17.9. One can study "products" <strong>of</strong> varieties in a sense which originated in group<br />
theory (see H. Neumann's book [330] ), and was later extended by A. I. Malcev (see<br />
[279, page 422]) to some other varieties. For recent developments, see [249], [258].