Deprecated: The each() function is deprecated. This message will be suppressed on further calls in /home/zhenxiangba/zhenxiangba.com/public_html/phproxy-improved-master/index.php on line 456 Working On It
Yet Another Paper on Group Theory Single Axioms
(it's only a temporary title)
First posted in June 2004, updated several times since.
This work (in progress) is toward showing that the there are no single axioms
shorter or simpler than those previously known for group theory in terms of
{product,inverse} and in terms of {division}.
An equation has size (length, vars), where
length is the number of occurrences of operations and variables
(including "="), and vars is the number of variables.
For example, x * e = x has size (5,1).
Mace4 was used
extensively to find couterexamples. A new option, the ability
to search for ring models, was added to Mace4 for this work,
and a new script was written to run a sequence of Mace4 jobs.
Abelian Groups
The answer for abelian groups is easy. The known
single axioms are the shortest possible.
Tarski's axiom for abelian groups in terms of
division has size (11,3) [Tarski-1938]
x / (y / (z / (x / y))) = z.
None can be simpler, because three variables are required, and
each variable must have an even number of occurrences.
For abelian groups in terms of product and inverse, we have an
axiom of size (12,3) [McCune-1993]
((x * y) * z) * (x * z)' = y.
None can be simpler, because three variables are required,
each variable must have an even number of occurrences,
and there must be an occurrence of inverse.
Ken Kunen shows in [Kunen-1992]
that the only possibility for simpler axioms is size (18,3).
Below we look at all of
the size (18,3) candidates and eliminate all but a few of them.
Properties of Single Equational Axioms
These apply to groups and subvarieties (and many other theories).
One side of the equation must be a variable, say x.
If alpha=x is a single axiom, neither the leftmost nor
rightmost variable in alpha can be x.
Otherwise there can be projection models.
The axiom must have at least three variables. Otherwise there can
be nonassociative models.
If alpha=x is a single axiom in terms of product and
inverse, then mirror(alpha)=x is also a single axiom.
Therefore, we can ignore product/inverse candidates (delta * gamma) = x, with
size(delta) > size(gamma).
Summary. There are no group theory identities of size (11,3)
satisfying the basic constraints.
There are
71 group theory identies
of sizes (15,3) and (15,4) satisfying those contraints,
and all of them are elimintated
by a single
non-group structure of size 7
(which turns out to be the smallest nonassociative inverse loop).
Look here for details on how the
candidates were generated and how the counterexample was found.
All candidates of size (18,3) satisfying the basic
constraints were generated.
1,981,980 equations were generated, and 20,568 of those are
group identites. Mace4 was used in various ways to find
counterexamples, and
23 candidates remain.
Look here for details on how the
candidates were generated and how the counterexamples were found.