Re: proof of (-1).a= -a

"Dave L. Renfro" <renfr1dl@xxxxxxxxx> wrote:
>
> An issue that seems to be rarely mentioned
> in textbooks is what justifies adding the
> same element to both sides of an equation.

It's a consequence of a basic inference rule of
EQUATIONAL LOGIC, i.e. the rules stating that
equality is an equivalence relation (reflexive,
symmetric, transitive) along with the rules of

SUBSTITUTION p(x) = q(x) -> p(a) = q(a)

REPLACEMENT a = b -> p(a) = p(b)

Now simply apply replacement to p(x) = x + c
to obtain your desired inference.

Except for fields, all of the algebraic structures
one meets in most first courses in algebra have
equational axiomatizations, so they are governed
by equational logic. See [3] for more on fields.

In 1935 Birkhoff proved the completeness of
equational logic, i.e. if an equational statement
holds true in all models then it is provable
(the converse, soundness, is trivial). The proof
is easy (about a half-page long) and can be found
in most math. logic or model theory textbooks.
(As with set theory, most mathematicians practice
their logic in a "naive" manner, which is probably
why questions like the above arise).

Note, however, even though completeness guarantees the
*existence* of equational proofs for true statements,
it may be quite difficult to discover such a proof.
Below is an example extracted from one of my recent posts:
-----------------
The above example is a trivial application of the interplay between
syntax and semantics that is characteristic of Model Theory. A much
deeper application is the model-theoretic proof of Jacobson's theorem
that rings satisfying X^m = X are commutative. This proceeds by
a certain type of factorization which reduces the problem to the
(subdirectly) irreducible factors of the variety. These turn out
to be certain finite fields, which are commutative, as desired.

[A simpler example of such a model-theoretic proof is the truth table
method for Boolean algebras: here the only subdirectly irreducible
Boolean algebra is the two element algebra B2, so a Boolean equation
holds true in an arbitrary Boolean algebra iff it holds true in B2 ]

By completeness there must also exist a purely equational proof but
even for small m this is notoriously difficult, e.g. m = 3 [1].
It's only recently that such a general non-model-theoretic proof was
discovered by John Lawrence (as Stan Burris informed me). I don't
know if it has been published yet, but see their earlier work [2].

--Bill Dubuque

[1] http://groups-beta.google.com/group/sci.math/msg/9b884af731351f10
http://google.com/groups?selm=y8z64vfamcc.fsf@xxxxxxxxxxxxxxxxxxxx

[2] S. Burris and J. Lawrence, Term rewrite rules for finite fields.
International J. Algebra and Computation 1 (1991), 353-369.
http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/fields3.pdf

[3] http://google.com/groups?selm=y8zoe8gu7g0.fsf@xxxxxxxxxxxxxxxxxxxx
.